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\hypersetup{pdfauthor={Chris K.~Caldwell, Wilfrid Keller, Angela Reddick, and Yeng Xiong},%
pdftitle={The History of the Primality of One: A Selection of Sources},%
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pdfkeywords={prime numbers, definition of prime, primality, one, unity, monad},%
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\vskip 1cm {\LARGE\bf The History of the Primality of One:\\
\vskip .1in
A Selection of Sources} \vskip 1cm
\large Chris K.~Caldwell, Angela Reddick,\footnote{\label{footnote-label} An undergraduate student at the time we began this project. Partially supported by a University of Tennessee at Martin Undergraduate Research Grant.} and Yeng Xiong\textsuperscript{\ref{footnote-label}} \\
Department of Mathematics and Statistics\\
University of Tennessee at Martin \\
Martin, TN 38238, USA\\
\href{mailto:caldwell@utm.edu}{\tt caldwell@utm.edu}\\
\href{mailto:yenxion@ut.utm.edu}{\tt yenxion@ut.utm.edu}\\
\ \\
Wilfrid Keller\\
Universit\"at Hamburg\\
Hamburg \\
Germany\\
\href{mailto:keller@rrz.uni-hamburg.de}{\tt keller@rrz.uni-hamburg.de}\\
\end{center}
\vskip .2 in
\begin{abstract}
The way mathematicians have viewed the number one (unity, the monad) has changed throughout the years. Most of the early Greeks did not view one as a number, but rather as the origin, or generator, of number. Around the time of Simon Stevin (1548--1620), one (and zero) were first widely viewed as numbers. This created a period of confusion about whether or not the number one was prime. In this dynamic survey, we collect a cornucopia of sources which deal directly with the question ``what is the smallest prime?'' The goal is to create a source book for studying the history of the definition of prime, especially as applied to the number one.
\end{abstract}
\section{Introduction}
It seems that a question like ``what is the first prime?'' would have the simple and obvious answer ``two.'' This is the most common answer throughout history, and the only accepted answer among mathematicians today, but it was not always the only answer. Some ancient Greeks defined the primes as a subset of the odd numbers, so started the sequence of primes with three. Others (including John Pell, John Wallis and Edward Waring) started the sequence of primes with the number one. We summarize this history in our companion article \cite{CX2012}. There we point out, for example, that from the ancient Greeks to the time of Stevin one was not even considered a number, so no one would ask if it was prime. The goal of this work is more simple: to collect a list of references helpful in addressing questions about the smallest primes in general, and about the primality of unity in particular.
When selecting sources, we sought all that made the author's view clear. This is often difficult because of language and typographical barriers. It is also difficult because few addressed the question explicitly. For example, Gauss does not even define prime in his pivotal \textit{Disquisitiones Arithmeticae} \cite{Gauss1966}, but his view on the primality of one can be implied from his statement of the fundamental theorem of arithmetic.
%%% (We will list other examples in section \ref{section examples} below.) %%% We will do this in a later version
We must not confuse what most authors wrote with what they believed to be correct. Definitions often depend on context. Those who teach undergraduate mathematics courses will be very familiar with the example of $\log x$. This expression often represents the common (base 10) logarithm in a pre-calculus course, the multivalued inverse of the exponential in a complex variables course, and the single-valued natural (base $e$) logarithm in a real analysis course. These differences are not matters of belief, just of tradition and context. So it is not surprising that V.-A.\ Lebesgue and G.\ H.\ Hardy seem ambivalent to the primality of one in the list below.
Finally, we appreciate that the Journal of Integer Sequences has allowed this work to be stored as a dynamic survey. This means that it can be edited and updated as time continues. We would be glad to hear of any significant additions or corrections (especially when you can share images or scans of the original texts!).
\subsection{Notes about the table of sources}
\begin{itemize}
\item When possible, we tried to reproduce the language, spelling and typography\footnote{However, we made only the most minimal effort to preserve line breaks and white-space. For example, some early publications placed blank spaces both before \textit{and} after punctuation marks such as a colon; yet we normally used the modern spacing and put this space only after the colon.} of the original sources. These could help the reader better understand the quote.
\item The table's first column, titled `one,' contains `yes' when the author mentioned one as a prime number.
\item When ellipses are in the original quotes, we will use `\ldots'. If we are using ellipses to denote the omission of part of a quote, we will use `[\,\ldots]'.
\item Any date before 1200 is an approximation.
\end{itemize}
\eject
%%% Chris, note to self:
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\section{Sources}
\label{section matrix}
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\hline \tabtop one & who\,/\,year & \hspace*{10em}quote (or comment) \\
\hline \endhead \hline \endfoot
no & Plato \newline 400\textsc{bce} & Tar\'an writes \cite[p.~276]{Taran1981}: ``The Greeks generally, Plato and Aristotle included,
considered two to be the first prime number (cf. Plato, \textit{Republic} \oldstylenums{524} D \oldstylenums{7}, \textit{Parmenides} \oldstylenums{143} C-\oldstylenums{144} A, pp.~\oldstylenums{14}-\oldstylenums{15} \textit{supra}, Aristotle, \textit{Physics} \oldstylenums{207} B \oldstylenums{5}-\oldstylenums{8}, \oldstylenums{220} A \oldstylenums{27}, \textit{Metaphysics} \oldstylenums{1016} B \oldstylenums{17}-\oldstylenums{20}, \oldstylenums{1021} A \oldstylenums{12}-\oldstylenums{13}, \oldstylenums{1052} B \oldstylenums{20}-\oldstylenums{24}, \oldstylenums{1053} A \oldstylenums{27}-\oldstylenums{30}, \oldstylenums{1057} A \oldstylenums{3}-\oldstylenums{6}, \oldstylenums{1088} A \oldstylenums{4}-\oldstylenums{8}, Euclid, \textit{Elem.} VlI, Defs.~\oldstylenums{1}-\oldstylenums{2}); and so for them one is not a number (Aristotle is explicit about this and refers to it as a generally accepted notion [cf. p.~\oldstylenums{20}, note \oldstylenums{95} and p.~\oldstylenums{35} with note \oldstylenums{175}]; for some late thinkers who treat one as an odd number cf. Cherniss, \textit{Plutarch's Moralia}, vol.~XIII, I, p.~\oldstylenums{269}, n.~\textit{d}). Nor did the early Pythagoreans consider one to be a number, since in all probability they subscribed to the widespread notion that number is a collection of units (cf. Heath, \textit{Euclid's Elements}, II, p.~\oldstylenums{280}; Cherniss, \textit{Crit. Pres. Philos.,} pp.~\oldstylenums{387} and \oldstylenums{389}).'' \xskip \\
yes & Speusippus \newline 350\textsc{bce} & Tar\'an writes \cite[p.~276]{Taran1981}: ``Speusippus, then, is
exceptional among pre-Hellenistic thinkers in that he considers one to be the first prime number. And Heath, \textit{Hist. Gr. Math.}, I, pp.~\oldstylenums{69}-\oldstylenums{70}, followed by Ross, \textit{Aristotle's Physics}, p.~\oldstylenums{604}, and others, is mistaken when he
contends\newline % this is here to make Greek not break badly
that Chrysippus, who is said to have defined one as $\pi\lambda\tilde{\eta}\theta o\varsigma$
\raisebox{4pt}{\tiny `}\hspace{-0.4pt}\raisebox{3pt}{\tiny $'$}\hspace*{-1ex}$\varepsilon\nu$
(cf. Iamblichus, \textit{In Nicom. Introd. Arith.}, p.~II, \oldstylenums{8}-\oldstylenums{9} [Pistelli]), was the first to treat one as a number (cf. further p.~\oldstylenums{38}f. with note \oldstylenums{189} \textit{supra}).'' \xskip \\ % missing two Greek accents
no & Aristotle \newline 350\textsc{bce} & Heath says \cite[p.~73]{Heath1981} that ``Aristotle speaks of the dyad as `the only even number
which is prime' (Arist. Topics, $\Theta$. 2, 157 a 39). Also Tar\'an \cite[p.~20]{Taran1981} states Aristotle explicitly argues one is not a number (Metaphysics 1088 A 6-8), saying ``Aristotle never considers one to be a number and for him the first number is two.'' See also \cite[p.~276]{Taran1981}. \xskip \\
no & Euclid \newline 300\textsc{bce} & Heath notes \cite[p.~69]{Heath1981}: ``Euclid implies [one is not a number] when
he says that a unit is that by virtue of which each of existing things is called one, while a number is `the multitude made up of units,' [\,\ldots].'' On page 73, Heath mentions that Euclid includes two among the primes. \xskip\\
no & Theon of Smyrna \newline 100\textsc{bce} & Smith writes \cite[p.~20]{Smith1958}: ``Aristotle, Euclid, and Theon of Smyrna defined a
prime number as a number `measured by no number but an unit alone,' with slight variations of wording. Since unity was not considered as a number, it was frequently not mentioned. Iamblichus says that a prime number is also called `odd times odd,' which of course is not our idea of such a number. Other names were used, such as `euthymetric' and `rectilinear,' but they made little impression upon standard writers.\\
& & The name `prime number' contested for supremacy with `incomposite number' in the Middle Ages, Fibonacci
(1202) using the latter but saying that others preferred the former.'' \\
& & Heath states \cite[p.~73]{Heath1981} that Theon of Smyrna sees two as ``odd-\textit{like} without being
prime'' and cites ``Theon of Smyrna, p.\ 24.\ 7.'' \xskip \\
no & Nicomachus \newline 100 & ``The Unit then is perfect potentially but not actually, for taking it into the sum as the first
of the line I inspect it according to the formula to see what sort it is, and I find it to be prime and incomposite; for in very truth, not by participation like the others, but it is first of every number and the only incomposite'' \cite[p.~20]{Johnson1916}. \\
& & Smith writes \cite[p.~27]{Smith1958}: ``It is not probable that Nicomachus (\textit{c.}~100) intended to exclude unity
from the number field in general, but only from the domain of polygonal numbers. It may have been a misinterpretation of the passage of Nicomachus that led Boethius to add the great authority of his name to the view that one is not a number.'' \\
& & Tar\'an notes \cite[p.~276]{Taran1981}: ``For, if we started the number series with three (as some Neopythagoreans did [cf. e.g. Nicomachus,
\textit{Intr. Arith.} I, {\sc{ii}}], who consider prime number to be a property of odd number only [cf. Tar\'an, \textit{Asclepius on Nicomachus}, pp.~\oldstylenums{77}-\oldstylenums{78}, on I, \textbgreek{nh} and \textbgreek{ca}, with references]), then there would be in ten three prime numbers (\oldstylenums{3}, \oldstylenums{5}, \oldstylenums{7}) and five composite ones (\oldstylenums{4}, \oldstylenums{6}, \oldstylenums{8}, \oldstylenums{9}, \oldstylenums{10}).'' \\
& & Heath notes that Nicomachus defines primes and composites as subdivisions of the odds
\cite[p.~73]{Heath1981}, so two is not prime. Also ``According to Nicomachus {\vold 3} is the first prime number [\,\ldots]'' \cite[p.~285]{Euclid1956}. \xskip\\
--- & Iamblichus \newline 300 & Heath also notes \cite[p.~73]{Heath1981} that Iamblichus defines primes and composites as subdivisions of the
odds, so two is not prime. \xskip \\
no & Martianus Capella \newline 400\vspace{-2\baselineskip} & Stahl and Johnson \cite[pp.~285--286]{Stahl1992} translate Martianus Capella as follows:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``[743] We have briefly discussed the numbers comprising the first series, the deities assigned to them, and the virtues of each number.
I shall now briefly indicate the nature of number itself, what relations numbers bear to each other, and what forms they represent. A number is a collection of monads or a multiple proceeding from a monad and returning to a monad. There are four classes of integers: the first is called `even times even'; the second `odd times even'; the third `even times odd'; and the fourth `odd times odd'; these I shall discuss later.\\
& & [744] Numbers are called prime which can be divided by no number; they are seen to be not `divisible' by the monad
but `composed' of it: take, for example, the numbers five, seven, eleven, thirteen, seventeen, and others like them. No number can divide these numbers into integers. So they are called `prime,' since they arise from no number and are not divisible into equal portions. Arising in themselves, they beget other numbers from themselves, since even numbers are begotten from odd numbers, but an odd number cannot be begotten from even numbers. Therefore prime numbers must of necessity be regarded as beautiful. \\
& & [745] Let us consider all numbers of the first series according to the above classifications: the monad is not a number; the dyad is an even number; the
triad is a prime number, both in order and in properties; the tetrad belongs in the even times even class; the pentad is prime; the hexad belongs to the odd times even or even times odd (hence it is called perfect); the heptad is prime; the octad belongs to [\,\ldots]''\\
%even times even; the ennead belongs to odd times odd; and the decad, %even times odd. These classifications apply equally to higher series. The first series runs from the monad to the ennead; the second from the %decad to ninety; the third from 0ne hundred to nine hundred; ...
& & [The numbers [743], [744] and [745] are in the quoted text, numbering the paragraphs.] \xskip \\
no & Boethius \newline 500 & Masi notes \cite[pp.~89--95]{Masi1983} that Boethius (like Nicomachus), defines prime as a subdivision of the odds,
and starts his list of examples at three. \xskip \\
no & Cassiodorus \newline 550 & A prime number, notes \cite[p.~5]{Grant1974}, ``is one which can be divided by unity alone; for example,
3, 5, 7, 11, 13, 17, and the like.'' For him, prime is a subset of odd; perfect, abundant and deficient are all subsets of even \cite[pp.~181--182]{Jones1966}. \xskip \\
no & Isidore of Seville \newline 636\vspace{-\baselineskip} & In ``Etymologiarum sive Originum, Liber III: De mathematica'' Isidore
says (Grant's translation\footnote{There is a wonderful 1493 version of this text online at \url{http://tudigit.ulb.tu-darmstadt.de/show/inc-v-1/0039}.\vfill\ } \cite[pp.~4--5]{Grant1974}):
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Number is a multitude made up of units. For one is the seed of number but not number. [\,\ldots] Number is divided into even and odd. Even number is divided into the following: evenly even, evenly uneven, unevenly even and unevenly uneven. Odd number is divided into the following: prime and incomposite, composite, and a third intermediate class \textit{(mediocris)} which in a certain way is prime and incomposite but in another way secondary and composite. [\,\ldots] Simple [or prime] numbers are those which have no other part [or factor] except unity alone, as three has only a third, five only a fifth, seven only a seventh, for these have only one factor.'' \xskip \\
no & Al-Khwarizmi \newline 825 & ``Boetius (\textsc{ad} 475--524/525), who wrote % Is is indeed spelled Nichomachus in this text
the most influential book of mathematics during the Middle Ages, \textit{De Institutione Arithmetica Libri Duo}, following a personal restrictive interpretation of Nichomachus and affirmed that one is not a number. Even Arab mathematicians (e.g.\ Abu Ja'far Mohammed ibn Musa Al-Khowarizmi, \textit{c.}~\textsc{ad} 825) excluded unity from the number field. Rabbi ben Ezra (\textit{c.}~1095--\textit{ca.} 1167), instead in his \textit{Sefer ha-Echad} (Book of Unity) argued that one should be looked upon as a number. Only during the 16th century did authors begin to raise the question as to whether this exclusion of unity from the number field was not a trivial dispute (Petrus Ramus, 1515--1572), but Simon Stevin (\textit{c.}~1548--\textit{c.} 1620) argued that a part is of the same nature as the whole, and hence, that unity is a number.'' \cite[p.~812]{PRT2000}. \xskip \\
% the / before the word number may indicate a change to a new line of the source he is translating
no & al-Kind{\=\i} \newline 850\vspace{-\baselineskip} & After considering and rejecting the possibility of one being a number
al-Kind{\=\i} writes \cite[p.~102]{AlKindi1974}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Since, therefore, it is clear that one is not a number, the definition said of number shall then encompass /number fully, \textit{viz.}, that it is a magnitude (composed of) onenesses, a totality of onenesses, and a collection of onenesses. Two is, then, the first number.'' (He did see the number two ``as prime, if in a qualified way'' \cite[p.~181]{AlKindi1974}.) \xskip \\
\pdfbookmark[1]{1000 A.D.}{1000}\label{1000}%
no & Hugh of St.\ Victor \newline 1120 & ``Arithmetic has for its subject equal,
or even, number and unequal, or odd, number. Equal number is of three kinds: equally equal, equally unequal, and unequally equal. Unequal number, too, has three varieties: the first consists of numbers which are prime and incomposite; the second consists of numbers which are secondary and composite; the third consists of numbers which, when considered in themselves, are secondary and composite, but which, when one compares them with other numbers [to find a common factor or denominator],
are prime and incomposite.'' \cite[p.~56]{Grant1974} \xskip \\
--- & Rabbi ben Ezra \newline 1140 & Smith notes \cite[p.~27]{Smith1958}: ``One writer, Rabbi ben Ezra (\textit{c.}~\oldstylenums{1140}), seems,
however, to have approached the modern idea. In his \textit{Sefer ha-Echad (Book on Unity)} there are several passages in which he argues that one should be looked upon as a number.'' \\
& & On the other hand, M.\ Friedl\"ander \cite[p.~658]{Friedlander1896} notes: ``The book [Rabbi ben Ezra's \textit{Arithmetic}] opens with a
parallelism between the Universe and the numbers; there we have nine spheres and a being that is the beginning and source of all the spheres, and at the same time separate and different from the spheres. Similarly there are nine numbers, and a unit that is the foundation of all numbers but is itself no number.'' \xskip \\
--- & Fibonacci \newline 1202\vspace{-\baselineskip} & Smith quotes Fibonacci's \textit{Liber Abaci} I, 30, as follows \cite[p.\ 20]{Smith1958}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Nvmerorum quidam sunt incompositi, et sunt illi qui in arismetrica et in geometria primi appellantur. [\,\ldots] Arabes ipsos hasam appellant. Greci coris canon, nos autem sine regulis eos appellamus.'' Besides Fibonacci's preferred `incomposite,' `simple number' also seems common in the later periods. See also the 1857 copy of \textit{Liber Abaci} \cite[p.~30]{Leonardo1857}. \xskip \\
no & Prosdocimo \newline 1483\vspace{-\baselineskip} & Smith writes \cite[pp.~13--14]{SP1908}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``This rare work was written for the Latin schools, and is a good example, the first to appear in print, of the non-commercial algorisms of the fifteenth century. It follows `Bohectius' (Boethius) in defining number and in considering unity as not itself a number, as is seen in the facsimile of the first page.'' \xskip \\
--- & ------ & L.~L.\ Jackson \cite[p.~30]{Jackson1906}, writing about the teaching of mathematics in the sixteenth century, notes:
\vspace{0.5\baselineskip} % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``This difference of opinion as to the nature of unity was not new in the sixteenth century. The definition had puzzled the wise men of antiquity. Many Greek, Arabian, and Hindu writers had excluded unity from the list of numbers. But, perhaps, the chief reason for the general rejection of unity as a number by the arithmeticians of the Renaissance was the misinterpretation of Boethius's arithmetic. Nicomachus (c.~100 A.D.) in his $A\rho\iota\theta\mu\eta\tau\iota\kappa\eta\varsigma$ $\beta\iota\beta\lambda\iota\alpha$ $\delta\nu o$ had said that unity was not a polygonal number and Boethius's translation was supposed to say that unity was not a number. Even as late as 1634 Stevinus found it necessary to correct this popular error and explained it thus: $3 - 1=2$, hence $1$ is a number.'' \xskip \\
no & John of \newline Holywood \newline 1488 & ``Therfor sithen pe ledynge of vnyte in hym-self ones or twies nought
comethe but vnytes, Seithe Boice in Arsemetrike, that vnyte potencially is al nombre, and none in act. And vndirstonde wele also that betwix euery.'' The editor noted beside this section that ``Unity is not a number'' \cite[p.~47]{Steele1922}. \xskip \\
%%%% Missing many ligatures ... not sure all words are correct
\pdfbookmark[1]{1500 A.D.}{1500}\label{1500}%
no & P.\ Ciruelo \newline 1526\vspace{-\baselineskip} & Ciruelo states \cite[p.~15]{Ciruelo1526} that primes are a subset of the odds:
\vspace{0.5\baselineskip} % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold Numeri imparis= tres= sunt species= immediate qu{\ae} sunt, primus=,
secundus=, \& ad alterum primus=. Numerus= impar primus= est qui sola vnitate parte aliquota metiri potest,
vt. 3. 5. 7. idem\que{} incompositus= nominatur, \& ratio vtrius\que{} denominationis= est eadem : quia numeri imparis= nulla potest esse pars= aliquota pr{\ae}ter vnitatem, nisi illa etiam sit numerus= impar.}'' This source did not have page numbers, but this quote is on the 15th page. \xskip \\
no & J.\ K\"obel \newline 1537\vspace{-\baselineskip} & Menninger \cite[p.~20]{Menninger1992} quotes:
\vspace{0.5\baselineskip} % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Wherefrom thou understandest that 1 is no number / but it is
a generatrix / beginning / and foundation of all other numbers.'' \\
& & He also gives the original: \\
& & ``\textit{Darauss verstehstu das I. kein zal ist / sonder es ist ein gebererin / anfang / vnd fundament aller anderer zalen.}'' \xskip \\
no & G.\ Zarlino \newline 1561\vspace{-\baselineskip} & Gioseffo Zarlino's influential music theory text \cite[p.~22]{Zarlino1561} says:
\vspace{0.5\baselineskip} % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold \textit{Li numeri Primi \& incomposti sono quelli, i quali non possono esser numerati o diuisi da altro numero, che dall' vnit\`a; come 2. 3. 5. 7. 11. 13. 17. 19. \& altri simili}}.'' \xskip \\
--- & S.\ Stevin \newline 1585 & Menninger writes \cite[p.~20]{Menninger1992}: ``Stevin, the man who first introduced the algorism of decimal fractions, was probably the first mathematician expressly to assert (in 1585) the numerical nature of One''. However, there are others. First might be Speusippus (ca.~365\textsc{bce}) \cite[pp.~264, 276]{Taran1981}, but these exceptions are rare and had little effect on common thought. Speusippus viewed one as prime. \xskip\\
\pdfbookmark[1]{1600 A.D.}{1600}\label{1600}%
no & P.~A.\ Cataldi \newline 1603 & Cataldi's treatise on perfect numbers \cite[pp.~28--40]{Cataldi1603} contains a factor table to 750 and
a list of primes below 750 (from 2 to 741). \xskip\\
no & C.\ Clavius \newline 1611\vspace{-\baselineskip} & Clavius, commenting on Euclid, wrote \cite[p.~307]{Clavius1611}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold PRIMVS numerus= est, quem
vnitas= sola metitur. \textit{Q V O D si numerum quempiam nullus= numerus=, sed sola vnitas= metiatur, it a vt neg,
pariter par, neg, pariter impar, neque impariter impar po{\ss}it dui, appellabitur numerus= primus=;
quales= sunt omnes= isti 2. 3. 5. 7. 11. 13. 17. 19. 23. 29. 31. \&c. Nam eos= sola vnitas= metitur.}}'' \xskip \\
%%% For both of these Henrion quotes the font is not quite right. The first is missing ligatures, both use differet \& symbols
no & D.\ Henrion \newline 1615\vspace{-\baselineskip} & Henrion \cite[p.~207]{Euclid1615}, expounding on Euclid's definition, wrote:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold 11. Nombre premier, est celuy qui est mesur\'e par la seule vnit\'e.\vspace{2pt}
\textit{C'est \`a dire, que si vn nombre n'est mesur\'e par aucun autre nombre, mais= seulement par l'vnit\'e, il est nombre premier, \& tels= sont tous= ceux-cy} 2.3.5.7.11.13.17.19.23.29.31. \textit{\&c. Car la seule vnit\'e mesure iceux.}}'' \\
& & This was very slightly reworded in a later 1676 edition \cite[p.~381]{Euclid1676} (after his death):\\
& & ``{\vold 11. Nombre premier, est celuy qui est mesur\'e par la seule unit\'e.\vspace{2pt}
\textit{C'est-\`a-dire, que si un nombre n'est mesur\'e par aucun autre nombre, mais= seulement par soy m\^eme, \& l'unit\'e, il est nombre premier, \& tels= sont tous= ceux-cy} 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, \textit{\&c. Car chacun d'iceux n'est mesur\'e par aucun autre nombre, mais= par la seule unit\'e.}''} \\
& & (Note that Denis Henrion and Pierre H{\'e}rigone are \textit{both} pseudonyms for the Baron Cl{\'e}ment Cyriaque de Mangin
(1580--1643), see the entry ``Pierre H\'erigone'' in MacTutor \cite{MacTutor}.) \xskip \\
no & M.\ Mersenne \newline 1625 & %% problem, cannot get the descending iltalic z to match the original \raisebox{-0.5ex}{\includegraphics[height=1.5ex]{long_z}}
``{\vold\textit{Les= nombres= premiers= entr'eus= sont ceus= qui ont la seule vnit\'e pour leur mesure commune : \&
les= nombres= composez sont ceux qui sont mesurez par quelque nombre, qui leur sert de mesure commune.}} \\
& & {\vold Ce Thor\^eme comprend la 13.~\& 14.~definition du 7, \& n'a besoin que d'explication: ie di donc premierementque le n\~obre
premier n'a autre mesure que l'vnit\'e, tel qu'est, 2, 3, 5 \&c. vous= treuuerez les= autres= nombres= premiers= par l'ordre naturel des= n\~obres=
\textit{impairs=}, si vous= en ostez tous= les= n\~obres= qui sont \'eloignez par 3.~nombres= du 3, % modern: trouverez
\& par cinq nombres= du 5, \& par 7, nombres= du 7, \& ainsi des= autres=, } [\,\ldots]'' \cite[pp.~298--299]{Mersenne1625}. \\
& & In another text Mersenne wrote: ``[\,\ldots] \vold{il faut multiplier tous= les= nombres= premiers= moindres= que 10,
a s=cauoir 2, 3, 5, 7.}'' \cite[p.~23]{Mersenne1983} \xskip \\
no & A.\ Metius \newline 1640 & %% in veryoldstyle s= is round s
``{\vold Numeri considerantur aut absolut\`e pe se : aut inter se relativ\`e. Numerus= absolut\`e pe se consideratus=, est aut per se Primus=,
aut Compositus=. Numerus= per se primus= est, quem pr\ae ter unitatem nullus= alius= numerus= metitur, \textit{quales= sunt 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, \&c. namque eos= sola unitas= dividit, ut nihil supersit.}}'' \cite[pp.~43--44]{Metius1640} \xskip \\
no & M.\ Bettini \newline 1642\vspace{-\baselineskip} &
Bettini \cite[p.~36]{Bettini1642} writes about Euclid's definition:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold Qui lib.~7 def.~11. sic: \textit{Primus= numerus= est, quem vnitas= sola metitur,} quales= sunt 2. 3. 5. 7. 11. 13. 17. 19. 23. 29. 31. \&c. primos= numeros= \& inuenire, \& infinitos= esse docet lib.~9. propos.~20.}'' \xskip \\
\pdfbookmark[1]{1650 A.D.}{1650}\label{1650}%
no & L\'eon de Saint-Jean \newline 1657 & ``{\vold Sunt insuper numeri \textit{Primi}, qui
sola vnitate, nec alio pr{\ae}ter vnitatem numero, mensurantur.
Dicitur autem numerus= vnus= alterum mensurare, qui multoties= repetitus= alterum ita explet ; vt nihil
superfluat, aut desit. Itaque vocantur numeri primi ac \textit{Simplices=}, quales= sunt 2. 3. 5. 7.
11. 1. 3.}[\textit{sic}] {\vold 17. 19. 23 \&c.}'' \cite[p.~581]{Leon1657} \xskip \\
no & F.\ v.\ Schooten \newline 1657 & Schooten includes a table of primes \cite[pp.~393--403]{Schooten1657} below 10,000 entitled ``Sectio V.
\textit{Syllabus numerorum primorum, qui continentur in decem prioribus chiliadibus.}'' This list of primes begins with two (p.~394). \xskip \\
yes & T.\ Brancker \& D.\ Pell \newline 1668\vspace{-\baselineskip}
& The introduction to Brancker's table of primes \cite[p.~201]{Rahn1668} describes the table and one of
its most common uses:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold It may be of great use sometimes= to have \textit{a complete and orderly enumeration of all incomposits= between 0, and 100,000, without
any mixture of Composits=}; thus= 1. 2. 3. 5. 7. 11. 13. \&c, leaving out 9, 21 and all other composits=.} [\,\ldots] \\
& & {\vold If to \emph{each of these primes=} you set the Briggian Logarithm, you may find the Logarithms= for \emph{all of the rest of the numbers=}
in the first 100 Chilliads=, by addition of the logarithms= of their incomposite Factors=.}'' \\
& & Maseres reprints the appendix from \emph{Teutsche Algebra} which contains the tables (see \cite[Preface p.~vii, 353]{Maseres1795}) as pages 353 to
416 of his text. \\
& & Bullynck states that ``Pell was involved in reading, correcting and supplementing the translation [Brancker's translation of Rahn's
{\it Teutsche Algebra}]; in the end he replaced almost half of Rahn's text with his own [\,\ldots] Brancker calculated the factor table afresh up to 100,000, following Pell's directions. After the English translation in 1668, the book would be generally known as Pell's {\it Algebra}, and the {\it Table of Incomposits} as Pell's {\it Table}, although [Balthasar] Keller and Brancker, independently, had calculated the table, and Rahn wrote the original work'' \cite[p.\ 143]{Bullynck2010}. \xskip \\
no & S.\ Morland \newline 1673 & ``{\vold A prime \textit{number} is= that which is= measured onely by
an Unite. That is= to say 2, 5, 7, 11, 13, \textit{\&c} are \textit{prime numbers=}, because neither of them can possibly be divided into equal parts= by any thing less= then an Unite.}'' \cite[p.~25]{Morland1673} \\
& & [Surely the exclusion of 3 from his list of primes was an accident.] \xskip \\
no & J.\ Moxon \newline 1679 & Moxon wrote the first English language dictionary of mathematics (which defines number on page 97, primes on page 118, and unity on page 162).\\
& & ``{\vold {\gothfamily \large Prime}, or {\gothfamily \large First Number}, Is= defined by \textit{Euclid} to be that which onely
Unity doth measure, as= 2, 3, 5, 7, 11, 13, 17, 19, 33 [\textit{sic}], 29, 31, \&c. for onely Unity can measure these.}'' \cite[p.~118]{Moxon1679}\\
& & ``{\vold {\gothfamily \large Number}, Is= commonly defined to be, \textit{A Collection of Units=}, or \textit{Multitude composed of Units=};
so that \textit{One} cannot be properly termed a \textit{Number}, but the begining of \textit{Number}: Yet I confess= this= (though generally received) to some seems= questionable, for against it thus= one might argue: A Part is= of the same matter of which is= its= Whole; An Unit is= part of a multitude of Units=; Therefore an Unit is= of the same matter with a multitude of Units=: But the matter and substance of Units= is= Number; Therefore the matter of an Unit is= Number. Or thus=, A Number being given, If from the same we subtract 0, (no Number) the Number given doth remain: Let 3 be the Number given, and from the same be taken 1, or an Unit, (which, as= these will say, is= no Number) then the Number given doth remain, that is= to say, 3, which to say, is absurd. But this= by the by, and with submission to better Judgments=.}'' \cite[p.~97]{Moxon1679} \xskip \\
% Does cioe have an accent, or is it noise on the image?
no & V.\ Giordano \newline 1680 &
``\,\textit{I}\hspace{-1.5pt}\begin{footnotesize}\begin{rotate}{80}\reflectbox{$\smallint\hspace*{5.1pt}$}%
\end{rotate}\end{footnotesize}\,\textit{{\vold uttii numeri, che non possono essere misurati giustamente da altri numeri,
cioe che non sono numeri parimente pari, ne parimente dispari, ne meno disparimente dispari, m\`a che possono essere misurati solamente dall'vnit\`a, si dicono numeri primi: come sono i seguenti 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 \&c.''}} \cite[p.~310]{Giordani1680} \xskip \\
no & G.\ Clerke \newline 1682 & ``{\vold Docuit \textit{Euclides=}, lib.
7. Definit.~11. numerum illum esse primum quem unitas= sola metitur,
hoc est, dividit, ita, 2 3 5 7 11 13 17 19 23 29 31, \&c. sunt omnes= primi:} [\,\ldots]'' \cite[p.~39]{Clerke1682}. \xskip \\
yes & J.\ Wallis \newline 1685\vspace{-\baselineskip} & Wallis' ``A Discourse of Combinations, Alternations, and Aliquot Parts'' is reprinted as pp.~269--352 of \cite{Maseres1795}.
Here he makes the following definitions \cite[p.~292]{Maseres1795} (see also \cite[p.~496]{Wallis1693}):
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold 6. It is= manifest that the Number 1, hath no
Aliquot Part, and but one Divisor, that is= 1. Because there is= no Number less= than itself that may be a part of it : But it measures= itself ; and therefore is= its= own Divisor.}\\
& & {\vold 7. Any other Prime Number hath one Aliquot Part, and Two Divisors=. For a \textit{Prime Number}, we call,
such as= is= measured (beside itself) by no other Number but an Unit. As= 2, 3, 5, 7, 11, \&c. Each of which are measured by 1, and by itself; but not by any other Number. And hath therefore 2 Divisors=, and 1 Aliquot Part; but no more.} \\
& & {\vold 8. Every \textit{Power} of a \textit{Prime Number} (other than of 1, which here is= understood to be excluded,) hath so many Aliquot
Parts= as= are the dimensions= of such Power; and one Divisor more than so. [\,\ldots]}''\xskip\\
no & T.\ Corneille \newline 1685\vspace{-\baselineskip} & Corneille, in his encyclopedic dictionary, defined \cite[p.~110]{Corneille1694}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold NOMBRE. s.m. \textit{Plusieurs= unitez % The long z or descending z is used here
consider\'ees= ensemble} [\,\ldots] \textit{Nombre premier,} Celuy que la seule unit\'e mesure;
comme 2. 3. 5. 7. 11. qu'on ne s\c{c}auroit mesurer par aucun autre nombre, [\,\ldots]}''. \xskip \\
yes & J.\ Prestet \newline 1689 & ``{\vold Je nommerai \textit{nombres= simples=} ou \textit{premiers=}, ceux qu'on ne peut diviser au
juste ou sans= reste par aucun autre entier que par eux-m\^emes= ou par l'unit\'e; comme chacun des= dix 1, 2, 3, 5, 7, 11, 13, 17, 19, 23.}''
\cite[p.~141]{Prestet1689} \xskip \\
no & C.\ F.\ M.\ Dechales \newline 1690\vspace{-\baselineskip} & Expounding on Euclid's book 7, Dechales writes \cite[p.~169]{Chales1690}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold 1. Unitas= est secund\`um quam unumquodque dicitur unum. Nempe ab unitate dicitur unus= homo, unus= leo, unus= lapis=. H{\ae}c definitio dat primam tantum unitatis= cognitionem, quod in pr{\ae}senti materia sussicit, unitatem enim per se melius= cognoscimus=, qu\`am ex quacumque definitione.} \\
& & {\vold 2. Numerus= est ex unitatibus= composita multitudo. \textit{Unde tot
habet partes= quot unitates=, denominationemque habet ex multitudine unitatum. Ex quo sequitur omnes= numeros= inter
se commensurabiles= esse, cum eos= unitas= metiatur.}}\\
& & {\vold 11. Primus= numerus= absolut\`e dicitur is= quem sola unitas= metitur, \textit{ut 2, 3. 5. 7. 11. 13, quia
nullam habent partem aliquotam unitate majorem.}}'' \xskip \\
no & A.\ Arnauld \newline 1690 & ``{\vold On dit qu'un nombre est nombre premier,
quand il n'a de mesure que l'unit\'e \& soy-m\^eme, (ce qui se sous=-entend sans= qu'on le dise.) Comme 2. 3. 5. 7. 11. 13, \&c.}''
\cite[p.~98]{Arnauld1690}. \xskip \\
no & J.\ Ozanam \newline 1691\vspace{-\baselineskip} & Ozanam, essentially an expositor, defines \cite[p.~27]{Ozanam1691}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold Le \textit{Nombre Premier} est celuy qui n'est mesur\'e par aucun nombre que par l'unit\'e: comme 2, 3, 5, 7, 11, 17, 19, \&c. On le nomme aussi \textit{Nombre lineaire}, \& encore \textit{Nombre incompos\'e}, pour le differencier du \textit{Nombre compos\'e}.}'' [Where is 13?] \xskip \\
--- & ------ & A{\u{g}}arg{\"u}n and {\"O}zkan, in ``A historical survey of the fundamental theorem of arithmetic''
\cite{AO2001} address the development of the fundamental theorem of arithmetic and affirm with C.~Goldstein \cite{Goldstein1992} that up to the 17th century mathematicians were not interested in the prime factorization integers for its own sake, but as a means of finding divisors. Note how this may alter the way you view the primality of one.\xskip\\
\pdfbookmark[1]{1700 A.D.}{1700}\label{1700}%
no & E.\ Phillips \newline 1720 & ``{\vold {\gothfamily \large Prime, Simple}, or {\gothfamily \large
Incomposit Number}, (in \textit{Arithm.}) is= a Number, which can only be measur'd or divided by it self, or by Unity, without leaving any Remainder; as= 2, 3, 5, 7, 11, 13, \textit{\&c}. are Prime Numbers=.}\\
& & {\vold {\gothfamily \large Composite} or {\gothfamily \large Compound Number}, is= that which may be divided by some
Number, less= than the Composite it self, but greater than Unity; as= 4,6, 8, 9, 10, \&c.}'' \cite[p.~460]{Phillips1720}. (This book does not have page numbers but this is on the 460th page.) \xskip \\
no & ``Shuli Jingyun'' \newline c.\ 1720& Denis Roegel \cite{Roegel2011} reconstructed the tables from the Siku Quanshu (c.\ 1782)
\textit{which are supposedly} copies of those from the Shuli Jingyun (1713-1723) \cite{Roegel2011}. The list of primes begins $2, 3, 5, 7, \ldots$\footnote{Original image \url{http://www.archive.org/stream/06076320.cn\#page/n66/mode/2up}.\vfill\ } \xskip \\
yes & J.\ Harris \newline 1723\vspace{-\baselineskip} & John Harris' dictionary \cite{Harris1723} defines:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold INCOMPOSITE \emph{Numbers=}, are the same with those \emph{Euclid} calls= \emph{Prime Numbers=}. In Dr. \emph{Pell's=} Edition of of \emph{Brancker's= Algebra}, there is= a Table, as= it's= there called, of \textit{Incomposite Numbers=}, less= than 100000; tho' it contains= far more \textit{Composite} than \textit{Incomposite Numbers=} [\,\ldots] `Tis= true that 2 and 5 are Incomposite Numbers=, as= well as= 1 and 3; but they are not put into the Tables=, because no other Incomposite Numbers= can terminate in them: [\,\ldots]}.'' \xskip \\
%%%% This one is missing a couple long-s i ligatures
no & F.\ Brunot \newline 1723 & ``{\vold\textit{Le Nombre entier} signifie une ou plusieurs= unitez de m\^{e}me genre lorsque l'on n'y
considere aucune partie}.'' \cite[p.~2]{Brunot1723} \\
& & ``{\vold\textit{Le Nombre premier, simple,} ou \textit{qui n'est pas= compos\'e,} est celui qui n'a aucunes= parties= aliquotes=
que l'unit\'e, comme 2, 3, 5, 7, 11, 13, \&c.}'' \cite[p.~3]{Brunot1723} \xskip \\
no & J.\ Cort\`es \newline 1724\vspace{-\baselineskip} & Cort\`es states that he follows Euclid on a previous page.
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold El Numero primero se dize aquel que de sola la unidad puede ser medido, y no de otro numero, como 2. 3. 5. 7. 11. 13. y otros= de esta manera.}'' \cite[p.~7]{Cortes1724} \xskip \\
no & E.\ Stone \newline 1726 & Edmund Stone's mathematical dictionary \cite[p.~293]{Stone1726} states: ``{\vold \textsc{Prime Numbers}, in Arithmetick, are those made only
by Addition, or the Collection of Unites=, and not by Multiplication : So an Unite only can measure it; as= 2, 3, 4, 5, \textit{\&c.} and is= by some call'd a \textit{Simple}, and by others= an \textit{Uncompound Number}.}'' (This book does not contain any page numbers but this is on the 293rd page.) \xskip \\
no & E.\ Chambers \newline 1728 & ``{\vold {\sc Prime} \textit{Number}, in Arithmetic, a Number
which can only be measur'd by Unity; or whereof 1 is= the only aliquot part. See {\sc Number}. Such are 5, 7, 11, 13, \textit{\&c}.}'' \cite[p.~871]{Chambers1728} \\
& & ``{\vold 'Tis= disputed among Mathematicians=, whether or no \textit{Unity}
be a \textit{Number}.---The generality of Authors= hold the Negative; and make \textit{Unity} to be only inceptive of Number, or the Principle thereof; as= a Point is= of Magnitude, and \textit{Unison} of Concord.} \\
& & {\vold \textit{Stevinus=} is= very angry with the Maintainers= of this= Opinion : and yet, if Number be defin'd a Multitude of \textit{Unites=}
join'd together, as= many Authors= define it, 'tis= evident \textit{Unity} is= not a Number.}'' \cite[p.~323] {Chambers1728} \xskip \\
no & J.\ Kirkby \newline 1735 & ``{\vold 53.~An \textit{Even Number} is= that which is= measured by 2. \newline
54.~An \textit{Odd Number} is= one more than an even Number. \newline 55.~A \textit{Prime} or \textit{Incomposite Number} is= that which no Number measures= but Unity, as= 3, 5, 7, 11, 13, 17, 19.}'' \cite[p.~7]{Kirkby1735} \xskip \\
no & C.\ R.\ Reyneau \newline 1739 & ``{\vold On remarquera sur les= nombres= que leurs=
diviseurs= premiers= ne sont pas= toujours= de suite les= nombres= premiers= 2, 3, 5, 7, 11, \textit{\&c}.}'' \cite[p.~248]{Reyneau1739} \xskip \\
yes & C.\ Goldbach \newline 1742 & Goldbach's letter to Euler \cite{Goldbach1742} (with what Euler would modify to the ``Goldbach Conjecture'') uses 1 as a prime in sums such as:
{ \footnotesize
$$ 4 = \left\{ \begin{array}{l}
1+1+1+1 \\
1+1+2 \\
1+3,
\end{array}\right.
\hfill
5 = \left\{ \begin{array}{l}
2+3 \\
1+1+3 \\
1+1+1+2 \\
1+1+1+1+1,
\end{array}\right.
\hfill
6 = \left\{ \begin{array}{l}
1+5 \\
1+2+3 \\
1+1+1+3 \\
1+1+1+1+2 \\
1+1+1+1+1+1.
\end{array}\right.\vspace{-2.5ex}$$}
\xskip \\
yes & G.\ S.\ Kr{\"u}ger \newline 1746 & Kr{\"u}ger's list of primes \cite[p.~839]{Kruger1746} (calculated by Peter Jaeger \cite[footnote, p.\ 15]{Euler1770a})
starts with 1 and ends with 100,999. \xskip \\
\pdfbookmark[1]{1750 A.D.}{1750}\label{1750}%
yes & M.\ L.\ Willig \newline 1759 & Willig's factor list (from 1 to 10,000) \cite[p.~831]{Willich1759} starts:
\begin{center}
\begin{tabular}{r|l}
\textbf{\small{I}} & {\frakfamily\large\fraklines Primz.}\\
\textbf{2} & {\frakfamily\large\fraklines Primz.}
\end{tabular}
\end{center} \xskip \\
yes & N.\ de la Caille \newline 1762& ``{\vold Numerus=, qui nullius= alterius=, quam unitatis=, est
multiplus=, dicitur \textit{numerus= primus=}. Horum numerorum ampl{\ae} tabl{\ae} apud varios= scriptores= extant; en eos=, qui centenario sunt inferiores=: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,}'' \cite[p.~13]{CS1762}. \xskip \\
no & L.\ Euler \newline 1770\vspace{-\baselineskip} & Euler writes \cite[pp.~14--16]{Euler1770a}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``But, on the other hand, the numbers 2, 3, 5, 7, 11, 13, 17, \&c. cannot be represented in the same manner by factors, unless for that purpose we make use of unity, and
represent 2, for instance, by $1 \times 2$. But the numbers which are multiplied by 1 remaining the same, it is not proper to reckon unity as a factor.\\
& & All numbers, therefore, such as 2, 3, 5, 7, 11, 13, 17, \&c. which cannot be represented by factors, are
called \emph{simple}, or \emph{prime numbers}; whereas others, as 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, \&c. which may be represented by factors are called \emph{composite numbers}. \\
& & Simple or prime numbers deserve therefore particular attention, since they do not result from the multiplication of two or more numbers.
It is also particularly worthy of observation, that if we write these numbers in succession as they follow each other, thus, $$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, \&c.$$ we can trace no regular order; there increments being sometimes greater, sometimes less; and hitherto no one has been able to discover whether they follow any certain law or not.'' \xskip \\
yes & J.~H.\ \newline Lambert \newline 1770 & Table VI, Numeri Primi, begins with {\vold 1, 2, 3, 5, 7, 11,}
\ldots; repeated in the Latin version (same table number and page) \cite[p.~73]{Lambert1798}. \xskip \\
no & S.\ Horsley \newline 1772 & ``{\vold Hence it follows=, that all the Prime numbers=, except
the number 2, are included in the series= of odd numbers=, in their natural order, infinitely
extended; that is=, in the series= 3. 5. 7. 9. 11. 13. 15. [\,\ldots]}'' \cite[p.~332]{Horsley1772}. \xskip \\
yes & A.\ Felkel \newline 1776 & Felkel's Table A \cite{Felkel1776} (at the front of his factor table) lists the primes from 1 to 20353. \xskip \\
yes & E.\ Waring \newline 1782\vspace{-\baselineskip} & Waring writes \cite[p.~379]{Waring1782} (translated to English \cite[p.~362b]{Weeks1991}):
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\vold 1. Omnis= par numerus= constat e duobus= primis= numeris=, \& omnis= impair numerus= vel est primus= numerus=, vel constat e tribus= primis= numeris=, \&c.}'' (Every even number is the sum of two primes; every odd the sum of three.) \\
& & ``{\vold 3. Haud dantur tres= primi numeri in arithmetic\^a progressione,
quorum communis= differentia haud divisibilis= sit per numerum 6; ni 3 sit primus= terminus= arithmeticae seriei, in quo casu possunt esse tres= \& haud plures= termini ejusdem arithmetic{\ae} seriei primi numeri, \& quorum communis= differentia haud divisibilis= sit per 6: hic excipiantur du{\ae} arithmetic{\ae} series= 1, 2, 3 \& 1, 3, 5, 7.}'' (Here he is explaining there are only two arithmetic sequences of primes which do not have a common difference divisible by 6.) \\
& & Also \cite[p.~391]{Waring1788}
``[\,\ldots] {\vold adding the prime numbers= 1, 2, 3, 5, 7, 11, 13, 19, \&c.} [\,\ldots]'' [Where is 17?] \xskip \\
yes & A.~G.\ Rosell \newline 1785 & ``De este modo, 1, 2, 3, 5, 7, 11, 13, \&c. son n{\'u}meros primeros,
y 4, 6, 8, 9, 10, \&c. n{\'u}meros compuestos.'' \cite[p.~39]{Rosell1785} \xskip \\
% Not the correct digit 1 (which should look like I)
yes & A.\ B{\"u}rja \newline 1786 & ``{\frakfamily\large\fraklines Eine Primzahl oder einfache Zahl nennet man
diejenige die durch keine andere, sondern nur allein durch die Einheit und durch sich selbst gemessen wird. Z.\ E.\ }\textbf{\large\vold 1\comma{} 2\comma{} 3\comma{} 5\comma{} 7\comma{} 11\comma{} 13\comma{} 17} {\frakfamily\large\fraklines sind Primzahlen. Da{\ss} aber jede Zahl durch die Einheit und durch sich selbst gemessen wird, bedarf keines: Beweises:.}'' \cite[p.~45]{Burja1786} \xskip \\ % [Nice cover]
no & F.\ Meinert \newline 1789 & ``{\frakfamily\large\fraklines So sind} \textbf{\large\vold 2\comma{} 3\comma{} 5\comma{}
7\comma{} 11\comma{} 13} {\frakfamily\large\fraklines \etc. Primzahlen;} \textbf{\large\vold 4\comma{} 6\comma{} 8\comma} {\frakfamily\large\fraklines \etc. aber zusammenge\-set{z}\-te Zahlen.}'' \cite[p.~69]{Meinert1789} \xskip \\
\pdfbookmark[1]{1800 A.D.}{1800}\label{1800}%
no & C.~F.\ Gauss \newline 1801\vspace{-\baselineskip} & Gauss states and proves (for the first time) the uniqueness case of
the fundamental theorem of arithmetic:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``A composite number can be resolved into prime factors in only one way'' \cite{Gauss1966}. Euler (1770) assumed and Legendre (1798) proved the existence part of this theorem \cite{AO2001}. (Preset (1689) used, and al-F\={a}ris{\=\i} (ca.~1320) may have also proved, the existence part of this theorem.) Gauss' table had 168 primes below 1000 in \cite[p.~436]{Gauss1863} (including 1 as prime would give 169). \xskip \\
yes & A.~M.\ Chmel \newline 1807 & ``{\vold Numerus= integer praeter s=e ipsum et unitatem
nullum alium divisorem (mensuram) hacens=, dicitur \textit{simplex}, vel \textit{numerus= primus=}, (Primzahl). Numerns= autem talis=, qui praeter se ipsum et 1, adhuc unum vel plures= divisores= habet, vocatur \textit{compositus=}. \textit{Coroll}.~1. Numeri \textit{primi} sunt: 1, 2, 3, 5, 7, 11, 13, 17, 19 etc. \textit{Compositi}: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 etc.}'' \cite[p.~65]{Chmel1807}. \xskip \\
no & G.~S.\ Kl\"ugel \newline 1808 & ``{\frakfamily\Large\fraklines Primzahl,} {\frakfamily\large\fraklines einfache Zahl,} {\bf (numerus primus)}
{\frakfamily\large\fraklines ist eine solche, welche keine ganze Zahlen zu Factoren hat oder, welche nur von der Einheit allein gemessen wird, wie die Zahlen} \textbf{\large\vold 2\comma{} 3\comma{} 5\comma{} 7\comma{} 11\comma{} 13\comma{} 17\comma{} 19\comma{} 23\comma{} 29} {\frakfamily\large\fraklines u.~s.~f.}'' \cite[p.~892]{KGM1808}. \xskip \\
no & P.\ Barlow \newline 1811 & Barlow writes \cite[p.~54]{Barlow1811}: ``[\,\ldots] we have \ 2 \ 3 \ 5 \ 7 \ldots \ 97, which are all the prime numbers under 100.''
Also, in 1847: ``A \textit{prime number} is that which cannot be produced by the multiplication of any integral factors, or that cannot be divided into any equal integral parts greater than unity.'' \cite[p.~642]{Barlow1847}. \xskip \\
yes & J.~G.\ Garnier \newline 1818 & ``\textit{Lambert}, et tout r\'ecemment l'astronome \textit{Burkardt} ont
donn\'e des tables tr\`es-\'etendues de nombres premiers qui servent \`a la d\'ecomposition d'un nombre en ses facteurs
nombres premiers.'' \cite[p.~86]{Garnier1818} \\
& & This is followed with a table of primes, starting at 1, extending to 500. \xskip \\
yes & O.\ Gregory \newline 1825\vspace{-\baselineskip} & Gregory's low level ``Mathematics for practical men'' states (English \cite[pp.~44--45]{Gregory1825},
German \cite[pp.~40--42]{Gregory1828}):
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``1. A \textit{unit}, or \textit{unity}, is the representation of any thing considered individually, without regard to the parts of which it is composed.
2. An \textit{integer} is either a unit or an assemblage of units: and a \textit{fraction} is any part or parts of a unit. [\,\ldots]
4. One number is said to \textit{measure} another, when it divides it without leaving any remainder. [\,\ldots]
8. A \textit{prime number}, is that which can only be measured by 1, or unity.''
On the next page he lists the first twenty primes starting with 1. \xskip \\
yes & A.~M.\ \newline Legendre \newline 1830\vspace{-\baselineskip} & Presenting Euclid's argument there are infinitely many primes, he begins:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Car si la suite des nombres premiers {\vold 1.2.3.5.7.11}, etc. \'etait finie, et que $p$ f\^ut le dernier ou le plus grand de tous, [\,\ldots]'' \cite[p.~14]{Legendre1830}. (See also the quote for de Mondesir in 1877.) \xskip \\
yes & F.\ Minsinger \newline 1832 & Minsinger's school book \cite[pp.~36--37]{Minsinger1832} lists the 169 primes below 1000: from 1 to 997. \xskip \\
no & M.~Ohm \newline 1834 & ``{\frakfamily\large\fraklines Note: Die erstern Primzahlen sind der Reihe
nach: {\normalsize $\mathbf 2$, $\mathbf 3$, $\mathbf 5$, $\mathbf 7$, $\mathbf{11}$, $\mathbf{13}$, $\mathbf{17}$, $\mathbf{19}$, $\mathbf{23}$, $\mathbf{29}$,}} [\,\ldots]'' \cite[p.~140]{Ohm1834}. (Martin Ohm is a mathematician, the physicist's Georg Ohm's younger brother.) \xskip \\
no & A.\ Reynaud \newline 1835 & ``\textit{Un nombre est dit} {\sc premier}, \textit{lorsqu'il n'est
divisible que par lui-m\^eme et par l'unit\'e.} [\,\ldots] On trouve de cette mani\`ere que les nombres premiers sont, {\vold 2, 3, 5, 7, [\,\ldots] 607, etc.}'' \cite[pp.~48--49]{Reynaud1835}. \xskip \\
no & F.\ Lieber \newline et al. \newline 1840\vspace{-\baselineskip} & Encyclop{\ae}dia Americana \cite[p.~334]{LWB1840} states:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\sc Prime Numbers} are those which have no
divisors, or which cannot be divided into any number of equal integral parts, less than the number of units of which they are composed; such as 2, 3, 5, 7, 11, 13, 17, \&c.'' \xskip \\
no & R.~C.\ Smith \newline 1842\vspace{-\baselineskip} & This low level school book states \cite[p.~118]{Smith1842}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``A Prime Number is one that is divisible only by itself or unity, as 2, 3, 5, 7, 11, 13, 17, \&c.'' \xskip \\
no & J.\ Ozanam \newline et al. \newline 1844 & Ozanam's \textit{R\'ecr\'eations} (1694) was reworked by Jean-Etienne Montucla in 1778:
``who so greatly enlarged and improved the `Recreations' of Ozanam, that he may be said to have made the work his own'' \cite[p.~vi]{Ozanam1844}. The 3rd edition of Charles Hutton's English translation \cite[p.~16]{Ozanam1844} states: \\
& & ``A \textit{prime} number is that which has no other
divisor but unity.'' The table of primes from 1 to 10,000 (on the same page) starts at 2. \xskip \\
no & C.\ Beck \newline 1845 & ``Tous les nombres premiers depuis 1 jusqu'\`a 1000 sont contenus dans le tableau suivant:\\
& & 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, [\,\ldots] 997.''
\cite[p.\ 57]{Beck1845} \xskip \\
yes & J.~B.\ Weigl \newline 1848 & Weigl's school book \cite[p.~28]{Weigl1848} writes: ``{\frakfamily\large\fraklines Die ersten Primzahlen sind: 1, 2,
3, 5, 7, 11, 13, 15} [\textit{sic}] {\bf\ldots}''. \xskip \\ % \ldots are in the text
% This one seems to contradict itself
no? & J.\ Thomson \newline 1849 & ``All whole numbers are either prime or composite; a \textit{prime} number being that
which is not produced by the multiplication of other integers, while a \textit{composite} one is the product of two or more such
factors. Thus 2, 3, 5, 7, 11, \&c. are primes; while 4, 6, 8, 9, 10, \&c. are composite.'' \cite[p.~63]{Thomson1849} \xskip \\
\pdfbookmark[1]{1850 A.D.}{1850}\label{1850}%
yes & E.\ Hinkley \newline 1853 & This odd low-level text is based on the author's tables of primes and factorizations to 20,000; supplemented with Brancker's
table from 20,000 to 100,000. The preface \cite[p.~3]{Hinkley1853} states: \\
& & ``{\sc This} is the first book, made or published in the country, devoted exclusively to the subjects of \textit{prime numbers} and \textit{prime factors}.''
Then on page 7: ``The numbers 1, 2 and 3, are evidently prime numbers.'' \xskip \\
no & P.~L.\ \newline Chebyshev \newline 1854\vspace{-\baselineskip} & Chebyshev's {\it Collected Works} \cite[p.~51]{Chebyshev1899} reprints his
\textit{M\'emoire sur les nombres premiers} from 1854 which states:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Ce sont les questions sur la valeur num\'erique des s\'eries, dont les termes sont des fonctions des nombres premiers 2, 3, 5, 7, 11, 13, 17, etc.'' \xskip \\
no & C.~J.\ \newline Hargreave \newline 1854 & Glaisher, in his ``Factor Table for the Fourth Million'' (1879), discusses counts of primes by others
\cite[pp.~34--35]{Glaisher1879} as follows. (For Hargreave, Glaisher cites \cite[pp.\ 114--122]{Hargreave1854}.) \\
& & ``The results obtained by these writers do not agree. Thus in the case of 1,000,000 the number of primes is determined by Hargreave at 78,494,
by Meissel at 78,498, and by Piarron de Mondesir at 78,490. The true number, excluding unity, as counted from the Tables is 78,498, agreeing with Meissel's result. Hargreave and Meissel exclude unity in their determinations, but M.\ de Mondesir includes it. [\,\ldots] Legendre counted the number of primes in the first million as 78,493, which, as he included unity, is in error by 6 (see p.\ 30).'' \xskip \\
yes & A.\ Comte \newline 1854\vspace{-\baselineskip} & The philosopher and non-mathematician J.\ S.\ Mill \cite[p.~196]{Mill1865} writes:\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``But M.~Comte's puerile predilection for prime numbers almost passes belief. His reason is that they are the type of irreductibility: each of them is a kind of ultimate arithmetical fact. This, to any one who knows M.~Comte in his later aspects, is amply sufficient. Nothing can exceed his delight in anything which says to the human mind, Thus far shalt thou go and no farther. If prime numbers are precious, doubly prime numbers are doubly so; meaning those which are not only themselves prime numbers, but the number which marks their place in the series of prime numbers is a prime number. Still greater is the dignity of trebly prime numbers; when the number marking the place of this second number is also prime. The number thirteen fulfils these conditions: it is a prime number, it is the seventh prime number, and seven is the fifth prime number. Accordingly he has an outrageous partiality to the number thirteen. Though one of the most inconvenient of all small numbers, he insists on introducing it everywhere.'' \\
& & There is an example of this in Comte's \textit{System of Positive Polity} \cite[p.~420]{Comte1854}. \xskip \\
no & V.-A.\ Lebesgue \newline 1856\vspace{-\baselineskip} & ``[\,\ldots] on repr\'esentera la suite compl\`ete des nombres
premiers par $p_0 = 2,\, p_1 = 3,\, p_2 = 5,\ldots,\, p_{i-1},\, p_i,\, p_{i+1}.$'' \cite[p.\ 130]{Lebesgue1856}
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
Note that Victor-Am\'ed\'ee Lebesgue is a number theorist. He is unrelated to Henri Lebesgue (1875--1941) who worked with integration and measure theory. \xskip \\
yes & V.-A.\ Lebesgue \newline 1859 & ``[\,\ldots] les nombres premiers 1, 2, 3, 5, 7, 11, 13, [\,\ldots]'' \cite[p.~5]{Lebesgue1859} \xskip \\
yes & V.-A.\ Lebesgue \newline 1862 & ``On entend par \textit{diviseur} d'un nombre $n$ tout nombre qui s'y
trouve contenu une ou plusiers fois exactement; quel que soit $n$, les nombres \oldstylenums{1} et $n$ en sont diviseurs. Le nombre $n$ est \textit{premier} lorsqu'il n'a que ces deux diviseurs; il est \textit{compos\'e} dans le cas contraire. Les nombres \oldstylenums{1}, \oldstylenums{2}, 3, 5, \oldstylenums{7}, \oldstylenums{11}, \oldstylenums{1}3, \oldstylenums{17}, \oldstylenums{19},\ldots sont premiers;'' \cite[p.~10]{Lebesgue1862} \xskip \\
no & L.\ Dirichlet \newline 1863 & Dedekind compiled Dirichlet's lectures, \textit{Vorlesugen \"uber Zahlentheorie} \cite[p.~12]{Dirichlet1863},
a few years after Dedekind died. He wrote: \\
& & ``Da jede Zahl sowohl durch die Einheit,
als auch durch sich selbst theilbar ist, so hat jede Zahl -- die Einheit selbst ausgenommen -- mindestens zwei (positive) Divisoren. Jede Zahl nun, welche keine anderen als diese beiden Divisoren besitzt, heisst eine \textit{Primzahl (numerus primus)}; es ist zweckm\"assig, die Einheit nicht zu den Primzahlen zu rechnen, weil manche S\"atze \"uber Primzahlen nicht f\"ur die Zahl 1 g\"ultig bleiben.'' \\
& & This last part is ``It is convenient not to include unity among the primes, because many theorems about prime numbers do
not hold for the number 1'' \cite[p.~8]{Dirichlet1999}. The parenthetical ``(positive)'' was not in the 1863 edition, but added by the 1879 edition \cite[p.~12]{Dirichlet1879}.\\
& & Nice quote: ``Thus in a certain sense the prime numbers are the material from which all other numbers may be built''
\cite[p.~9]{Dirichlet1999}. \xskip \\
yes & J.\ Bertrand \newline 1863\vspace{-\baselineskip} & Joseph Bertrand defines primes (definition 109, \cite[p.~86]{Bertrand1863}) by:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Un nombre entier est dit premier lorsqu'il n'a pas d'autres diviseurs entiers que lui-m\^eme et l'unit\'e. \\
& & {\sc Exemples}. 2, 3, 5, 7, sont des nombres premiers,
9 n'est pas premier, car il est divisible par 3.''\\
& & Despite this example, Table I is titled ``Contenant tous les nombres premiers depuis 1 jusq'\`a 9907'', and starts, just like it says, at 1 (p.~342). \xskip \\
no & V.-A.\ Lebesgue \newline 1864 & The ``TABLEAU des nombres premiers impairs, inf\'erieurs \`a 5500'' lists 24 odd prime less than 100,
starting at 3 \cite[p.~12]{Lebesgue1864}. \xskip \\
yes & J.~Ray \newline 1866\vspace{-\baselineskip} & Joseph Ray's elementary algebra text states \cite[p.~50]{Ray1866}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``A \textbf{Prime Number} is one which has no divisor except itself and unity.\\
& & A \textbf{Composite Number} is one which has one or more divisors besides itself and unity. Hence, \\
& & All numbers are either prime or composite; and every composite number is the product of two or more prime numbers. \\
& & The prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, etc.'' \xskip \\
--- & C.\ Aschenborn \newline 1867\vspace{-\baselineskip} & Aschenborn's arithmetic text for artillery and engineering school \cite[p.~86]{Aschenborn1867} explains how to determine the least common multiple of two numbers and then states:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``{\frakfamily\large\fraklines Man thut dies: nach der Reihe mit den Primzahlen {\normalsize$\mathbf 2$}, {\normalsize$\mathbf 3$}, {\normalsize$\mathbf 5$},} {\frakfamily\large\fraklines .\,.\,.\,.\,, bis: je} {$\mathbf 2$} {\frakfamily\large\fraklines der gegebenen, resp.\ verkleinerten Zahlen relative Primzahlen sind.}'' \xskip \\
no & E.\ Meissel \newline 1870 & ``Es sei $p_1 = 2$; $p_2 = 3$; $p_3 = 5$; \ldots\,$p_n$ die $n^{\rm te}$ Primzahl; [\,\ldots]'' \cite[p.\ 636]{Meissel1870}. \xskip \\
yes & A.\ J.\ Manchester \newline 1870\vspace{-2\baselineskip} & A lesson \cite[p.~131]{RhodeIsland1870} in a periodical for Rhode Island school teachers instructs:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``A \textit{prime} number can be divided by no whole number, except 1 (one) and itself without a remainder. 2, 5, 17, 29, 47, 13 are prime numbers.\\
& & A composite number can be divided by some whole number besides itself, and 1 (one) without a remainder. 10, 21, 49, 51, 87, 39, 46 are composite numbers.\\
& & \textit{Teacher}. Name all of the prime numbers from 1 to 50.\\
& & \textit{Pupil}. 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.''\\
& & And on page 133: ``\textit{Teacher}. Name the composite numbers from 1 to 150 that, at first sight, seem to be prime.\\
& & \textit{Pupil}. 39, 51, 57, 67, 89, 91, 93, 111, 117, 119, 123, 129, 133, 141, 143, 147.'' [Surely ``67, 89'' was meant to be ``69, 87.''] \xskip \\
yes & E.\ Brooks \newline 1873\vspace{-\baselineskip} & Edward Brooks, in his schoolbook \cite[p.~58]{Brooks1873} which is mostly questions and few answers, wrote:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``Numbers which can be produced by multiplying together other numbers, each of which is greater than a unit, are called \emph{composite numbers}. \\
& & Numbers which cannot be produced by multiplying together two or more numbers, each of which is greater than a unit, are called \emph{prime numbers}.'' \xskip \\
no & G.\ M'Arthur \newline 1875 & In Encyclop{\ae}dia Britannica, 9th ed., the entry
for Arithmetic \cite[p.~528]{MArthur1898} states: ``A \textit{prime number} is a number which no other, except unity, divides without a remainder; as 2, 3, 5, 7, 11, 13, 17, \&c.'' \\
& & Later an example: ``The \textit{prime factors} of a number are the prime numbers of which it is the continued product.
Thus, 2, 3, 7 are the prime factors of 42; 2, 2, 3, 5, of 60.'' \xskip \\
yes & J.\ Glaisher \newline 1876 & ``\textsc{M.~Glaisher}, en comptant 1 et 2 comme premiers, a trouv\'e les valeurs suivantes: [\,\ldots]''
\cite[p.~232]{Lucas1878}. This is ``Mister Glaisher, by counting 1 and 2 as first, has found the following values: [\,\ldots]''. \\
& & Also, in an appendix of his ``Factor table for the Fourth Million'' (1879), Glaisher gives a list \cite[p.~48]{ Glaisher1879} of primes
from 1 to 30,341; a list which literally begins with unity. \xskip \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%% need a less ambiguous Weierstrass citation %%%%%%%%%%%%%%%%%%%%%%%%%%%
no & K.\ Weierstrass \newline 1876 & ``Dies f\"uhrt zu dem Begriff der Primzahlen. Nimmt man die
Prim\-zahlen s\"ammtlich als positiv an, so kann man jede Zahl als Product von Primzahlen und einer Einheit $+1$ oder $-1$ darstellen, und zwar auf eine einzige Weise. \\
& & Der Begriff der Primzahlen kann im Gebiete der complexen ganzen Zahlen, die aus den vier Einheiten $1,$ $-1,$ $i,$ $-i$
durch Addition zusammengesetzt sind, aufrecht erhalten werden. Denn jede Zahl $a+bi$ l\"asst sich auf eine einzige Weise durch ein Product von prim\"aren Primzahlen und einer der vier Einheiten ausdr\"ucken.'' \cite[p.~391]{Weierstrass1902} \xskip \\
yes & P.\ de Mondesir \newline 1877 & Glaisher, in his ``Factor Table for the Fourth Million'' (1879), discusses counts of primes by others
\cite[pp.~34--35]{Glaisher1879} as follows. (For Piarron de Mondesir, Glaisher cites \cite{Mondesir1878}.) \\
& & ``The results obtained by these writers do not agree. Thus in the case of 1,000,000 the number of primes is determined by Hargreave at 78,494,
by Meissel at 78,498, and by Piarron de Mondesir at 78,490. The true number, excluding unity, as counted from the Tables is 78,498, agreeing with Meissel's result. Hargreave and Meissel exclude unity in their determinations, but M.\ de Mondesir includes it. [\,\ldots] Legendre counted the number of primes in the first million as 78,493, which, as he included unity, is in error by 6 (see p.\ 30). \\
& & M.\ de Mondesir finds the number of primes inferior to 100,000 (including unity) to be 9,593, and remarks that Legendre gave 9,592. The former
value is the correct one.'' \xskip \\
no & H.\ Scheffler \newline 1880 & ``Hiernach sind die reellen Primzahlen 2, 3, 5, 7 \ldots, welche fr\"uher
daf\"ur gehalten wurden, s\"ammtlich gemeine reelle Primzahlen [\,\ldots]'' \cite[p.~79]{Scheffler1880}. \xskip \\
%%%%%%% Too vague???????? Add if you get a clear Burckhardt report.
%yes & 1880 & \textbf{G.~S.~Carr} & \cite{Carr1880} & `Prime number' is never defined and one is never mentioned in the
% context of primes or prime factors, however he
% reproduces Burckhardt's table in its entirety (Burckhardt's lists 1 as prime). In the table of contents has an
% entry: ``Common and Hyperbolic Logarithms of the Prime Numbers from 1 to 109'' (of course the table starts at 2).\\
no & G.\ Wertheim \newline 1887 & ``Wir wollen die Anzahl der Zahlen des Gebiets von 1 bis $n,$ welche durch
keine der $i$ ersten Primzahlen $p_1=2, p_2=3, p_3=5, \ldots , p_i$ theilbar sind, durch $\varphi(n,i)$ bezeichnen.'' \cite[p.~20]{Wertheim1887} \xskip \\
no & P.~L.\ \newline Chebyshev \newline 1889 & ``\textit{Einfach} heisst eine Zahl, welche nur durch \textit{Eins} und
durch sich selbst theilbar ist; eine solche wird auch \textit{Primzahl} genannt. \textit{Eine zusammengesetzte Zahl} nennt man dagegen eine solche, welche durch eine andere Zahl, die gr\"osser als \textit{Eins} ist, ohne Rest getheilt werden kann. So sind 2, 3, 5, 7, 11, und viele andere \textit{Primzahlen}, hingegen 4, 6, 8, 9, 10 und andere dergleichen \textit{zusammengesetzte Zahlen}.'' \cite[pp.~2--3]{Chebyshev1889} \xskip \\
yes & A.\ Cayley \newline 1890 & Encyclop{\ae}dia Britannica, 9th ed., entry for number \cite[p.~615]{Cayley1890}: ``In the ordinary
theory we have, in the first instance, positive integer numbers, the unit or unity 1, and the other numbers 2, 3, 4, 5, \&c. [\,\ldots].'' \\
& & ``A number such as 2, 3, 5, 7, 11, \&c., which is not a product of numbers, is said to be a prime number; and a
number which is not prime is said to be composite. A number other than zero is thus either prime or composite; [\,\ldots].'' \\
& & ``Some of these, 1, 2, 3, 5, 7, \&c. are prime, others, $4, =2^{2}, 6, =2.3,$ \&c., are composite; and we have the
fundamental theorem that a composite number is expressible, and that in one way only, as a product of prime factors, $N = a^{\alpha}b^{\beta}c^{\gamma} \ldots (a, b, c, \ldots$ primes other than 1; $\alpha, \beta, \gamma, \ldots$ positive integers).'' \xskip \\
no & E.\ Lucas \newline 1891 & ``Il y a donc deux esp\`eces d'entiers positifs, les nombres premiers et
les nombres compos\'es; mais on doit observer que l'unit\'e ne rentre dans aucune de ces deux esp\`eces et, dans la plupart des cas, il ne convient pas de consid\'erer l'unit\'e comme un nombre premier, parce que les propri\'et\'es des nombres premiers ne s'appliquent pas toujours au nombre 1.'' In a footnote he gives the example ``Ainsi le nombre 1 est premier \`a lui-m\^eme, tandis qu'un nombre premier $p$ n'est pas premier \`a lui-m\^eme; [\,\ldots]''
\cite[p.\ 350]{Lucas1891}. \xskip \\
yes & W.\ Milne \newline 1892 & This low-level school book is mostly questions, few answers. ``Thus 1, 3, 5, 7, 11, 13, etc.,
are prime numbers.'' \cite[p.~92]{Milne1892}. On page 95, 1 is not listed as a prime factor of 1008. \xskip \\
yes & R.\ Fricke \newline 1892 & ``Man bezeichne nun die Primzahlen 1, 2, 3, 5, \ldots'' \cite[p.~592]{Fricke1892}. \xskip \\
no & J.~P.\ Gram \newline 1893 & Gram reports $\pi(100000) = 9592$, which is true if 1 is omitted from the primes.
(He uses $\theta$ instead of $\pi$.) \cite[p.~312]{Gram1893}. \xskip \\
no & P.\ Bachmann \newline 1894 & ``Denkt man sich sodann alle Primzahlen bis zu einer
bestimmten Primzahl $p$ hin, 2, 3, 5, 7, \ldots $p_0,$ $p,$ so sei [\,\ldots]'' \cite[p.~135]{Bachmann1894}. \xskip \\
yes & R.\ Fricke \& F.\ Klein \newline 1897 & ``Die der Primzahl $x$ voraufgehenden Primzahlen seien $1, 2, 3, 5,
\ldots, \lambda$, so dass $l$ ein Multiplum des Productes $2\cdot 3\cdot 5\cdots \lambda$ ist.'' \cite[p.~609]{FK1897} \xskip \\
\pdfbookmark[1]{1900 A.D.}{1900}\label{1900}%
yes & L.\ Kronecker \newline 1901 & ``[\,\ldots] da{\ss} die 16 Primzahlen 1, 2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 kleiner sind als 50.'' \cite[p.~303]{Kronecker1901} \xskip \\
yes & G.\ Chrystal \newline 1904 & ``It is also obvious that every integer (other than unity) has at least two divisors, namely,
unity and itself; if it has more, it is called a \textit{composite integer}, if it has no more, a \textit{prime integer}. For example, 1, 2, 3, 5, 7, 11, 13, \ldots are all prime integers, whereas 4, 6, 8, 9, 10, 12, 14 are composite.'' \cite[p.~38]{Chrystal1904} \xskip \\
yes & G.~H.\ Hardy \newline 1908+ & Hardy's first-year university textbook \cite{Hardy1908} states that 1 is prime in at least two
places. First, while discussing Euclid's proof that there are infinitely many primes, Hardy notes
\cite[p.~122]{Hardy1908}\label{hardy1908}: \\
& & ``If there are only a finite number of primes let them be 1, 2, 3, 5, 7, 11, \ldots $N$.'' \\
& & This was unchanged for the first six editions of his text 1908, 1914, 1921 \cite[pp.\ 143--4]{Hardy1921}, 1925, 1928 and 1933.
(See the Hardy 1938 entry.) Next, he writes \cite[p.~147]{Hardy1908}: \\
& & \label{Hardy's example .111 ...}``The decimal $.111\,010\,100\,010\,10\ldots$,
in which the $n$th figure is $1$ if $n$ is prime, and zero otherwise, represents an irrational number.'' \\
& & This example remained the same in all 10 editions (e.g., \cite[p.~174]{Hardy1921}, and ``the revised 10th edition'' 2008 \cite[p.~151]{Hardy2008}).
He also has the ambiguous statement (\cite[p.~48]{Hardy1908}, \cite[p.~56]{Hardy1921}): \\
& & ``Let $y$ be defined as
\textit{the largest prime factor of $x$} (cf. Exs. VII. 6). Then $y$ is defined only for integral values of $x$. When
\begin{center}
$ \arraycolsep=2pt % narrow the space between columns
\begin{array}{rrrrrrrrrrrrrrrrr}
x & = & \pm & 1, & 2, & 3, & 4, & 5, & 6, & 7, & 8, & 9, & 10, & 11, & 12, & 13, & \ldots \\
y & = & & 1, & 2, & 3, & 2, & 5, & 3, & 7, & 2, & 3, & 5, & 11, & 3, & 13, & \ldots
\end{array}
$
\end{center}
The graph consists of a number of isolated points.''\\
& & This is essentially unchanged in the revised 10th edition \cite[p.~151]{Hardy2008}; but whether or not 1 is considered prime, it is
reasonable to accept 1 as the largest prime factor of 1. (Certainly 1 is the largest prime power dividing 1.) \xskip \\
no & E.\ Landau \newline 1909 & ``Unter einer Primzahl versteht man eine positive ganze Zahl, welche
von 1 verschieden und nur durch 1 und durch sich selbst teilbar ist. \\
& & Die Reihe der Primzahlen beginnt mit
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, \ldots .'' \cite[p.~3]{Landau1909}. \xskip \\
no & W.~F.\ Sheppard \newline 1910\vspace{-\baselineskip} & Encyclop{\ae}dia Britannica's (11th ed.) entry
for `arithmetic' \cite[p.~531]{Sheppard1910} states:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``A number (other than {\vold 1}) which has no factor except itself is called a \textit{prime number}, or, more briefly, a \textit{prime}.
Thus {\vold 2, 3, 5, 7} and {\vold 11} are primes, for each of these occurs twice only in the table. A number (other than {\vold 1}) which is not a prime number is called a \textit{composite} number.''\\
& & ``The number {\vold 1} is usually included amongst the primes; but, if this is done, the last paragraph [talking about the
fundamental theorem of arithmetic] requires modification, since {\vold 144} could be expressed as {\vold 1. 2}$^4${\vold. 3}$^2$, or as {\vold 1}$^2${\vold. 2}$^4${\vold. 3}$^2$, or as {\vold 1}$^p${\vold. 2}$^4${\vold. 3}$^2$, where $p$ might be anything.'' \xskip \\
no & G.~B.\ Mathews \newline 1910\vspace{-\baselineskip} & Encyclop{\ae}dia Britannica's (11th ed.) entry
for `number' \cite[p.~851]{Mathews1910} reads: \\
& & ``The first noteworthy classification of the natural numbers is into those which are prime and those which are composite. A prime number is
one which is not exactly divisible by any number except itself and {\vold 1}; all others are composite.''\\
& & That definition is ambiguous, but later on the same page to he clearly is excluding unity from the primes: \\
& & ``Every number
may be uniquely expressed as a product of prime factors.
\qquad Hence if $n = p^\alpha q^\beta r^\gamma \ldots$ is the representation of any number $n$ as the product of powers of different
primes, the divisors of $n$ are the terms of the product $(${\vold 1}$+p+p^2 + \ldots + p^\alpha)(${\vold 1}$+q+ \ldots + q^\beta) (${\vold 1}$+r+\ldots+r^\gamma)\ldots$ their number is $(\alpha + ${\vold 1}$)(\beta+${\vold 1}$)(\gamma+${\vold 1}$)\ldots,$ and their sum is $\Pi (p^{\alpha+1}{-}${\vold 1}$){\div}\Pi(p{-}${\vold 1}$)$.''\\
& & The same article \cite[p.~863]{Mathews1910} later states: ``Similar difficulties are encountered when we examine Mersenne's numbers, which are
those of the form $2^p-1$, with $p$ a prime; the known cases for which a Mersenne number is prime correspond to $p$ = {\vold 2, 3, 5, 7, 13, 17, 19, 31, 61.}'' If 1 was prime, then so would be $2^1-1$. \xskip \\
no & H.~v.\ Mangoldt \newline 1912 & ``Ein anderes Beispiel ist die Reihe \ $2; \ 3; \ 5; \ 7; \ 11; \ \cdots$ der Prim\-zahlen.'' \cite[p.~176]{Mangoldt1912} \xskip \\
yes & D.~N.\ Lehmer \newline 1914\vspace{-\baselineskip}& Lehmer begins the introduction to his \textit{List of Prime Numbers From} 1 \textit{to} 10,006,721 as follows \cite{lehmer1914}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``A prime number is defined as one that
is exactly divisible by no other number than itself and unity. The number 1 itself is to be considered as a prime according to this definition and has been listed as such in the table. Some mathematicians [a footnote here cites E.~Landau \cite{Landau1909}], however, prefer to exclude unity from the list of primes, thus obtaining a slight simplification in the statement of certain theorems. The same reasons would apply to exclude the number 2, which is the only even prime, and which appears as an exception in the statement of many theorems also. The number 1 is certainly not composite in the same sense as the number 6, and if it is ruled out of the list of primes it is necessary to create a particular class for this number alone.'' \xskip \\
no & E.\ Hecke \newline 1923 & ``Wenn es au{\ss}er der trivialen Zerf\"allung in ganzzahlige Faktoren, bei der ein Faktor $\pm$1 und der andere
$\pm a$ ist, keine andere gibt, so nennen wir $a$ eine \textbf{Primzahl}. Solche gibt es, wie $\pm 2, \pm 3, \pm 5$ \ldots. Die Einheiten $\pm 1$ wollen wir nicht zu den Primzahl rechnen.'' \cite[p.~5]{Hecke1923} \xskip \\
no & G.~H.\ Hardy \newline 1929 & ``More amusing examples are (c) $0.01101010001010 \cdots$ (in which the 1's have prime
rank) and (d) $0.23571113171923 \cdots$ (formed by writing down the prime numbers in order).'' \cite[p.~784]{Hardy1929} \\
& & Example (c) is in all the editions of his \textit{A Course of Pure Mathematics} where he included 1 as prime (so there it starts $0.111$).
This was never corrected in that text (see the Hardy 1908 entry). Here he begins Euclid's proof that there are infinitely primes as follows \cite[p.\ 802]{Hardy1929}: \\
& & ``If the theorem is false, we may denote the primes by $2, 3, 5, \cdots, P$, and all numbers are divisible by one of these.'' \xskip \\
no & G.~H.\ Hardy \newline 1938 & \label{hardy1938}In the seventh edition of his text \textit{A Course of Pure Mathematics}, Hardy starts Euclid's proof of the
infinitude of primes as follows \cite[p.~125]{Hardy1938}: \\
& & ``Let $2, 3, 5, \ldots, p_N$ be all the primes up to $p_N$, \ldots'' \\
& & This is a change from the the first six editions where unity was prime (see the Hardy 1908 entry). This new wording
was used from the 7th edition (1938) through the revised 10th edition (2008). \xskip \\
yes & M.\ Kraitchik \newline 1942 & Kraitchik's recreational mathematics text \cite[p.~78]{Kraitchik1953} says ``For example, there are 26 prime numbers between 0 and 100,
only 21 between 100 and 200, and no more than 4 between $10^{12}$ and $10^{12}+100$.'' \xskip \\
no & B.\ L.\ van der Waerden \newline 1949 & ``An element $p\neq 0$ which admits only trivial factorizations of the kind $p=ab$, where $a$ or $b$ is a unit,
is called an \textit{indecomposable element} or a \textit{prime element}. (In the case of integers we say: \textit{prime number};$^*$ in the case of polynomials: \textit{irreducible polynomial}.)'' \\ %
& & The footnote on `prime number' is: \\ % The footnote is actually numbered 12 in the text.
& & ``By prime numbers we usually understand only the positive prime numbers $\neq 1$, such as 2, 3, 5, 7, 11, \ldots'' \cite[p.~59]{vdWaerden1949}. \xskip \\
% no & 1962 & D.~Shanks & \cite[p.~3]{Shanks78} & ``If $p$ is an integer, $>1$, which is divisible only by $\pm 1$ and
% by $\pm p$, it is called \textit{prime}. And integer $>1$, not a prime, is called \textit{composite}.'' \\
yes & A.\ H.\ Beiler \newline 1964\vspace{-\baselineskip} & Beiler, a well-known expositor, wrote about the ``ubiquitous primes'' \cite[p.~211]{Beiler1964}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``From the humble 2, the only even prime, and 1, the smallest of the odd primes, they rise in an unending succession aloof and irrefrangible.'' [See also pp.~212--13, 223.] \xskip \\
no & J.\ Shallit \newline 1975 & Jeffrey Shallit \cite{Shallit1975}, as a student, wrote an interesting note about the prime factorization
of one suggesting that its prime factorization should be regarded as the empty list. \xskip \\
yes & C.\ Sagan \newline 1997 & The aliens in Carl Sagan's novel \textit{Contact} \cite[p.~76]{Sagan1997} transmit the first 261
primes starting with one: 1, 2, 3, 5, 7, \ldots, to signal their existence. \xskip \\
% Contacted this source, offered to correct the entry, sarcastic offer to rewrite the whole dictionary as a response (Oct 2012)
% ... are in the original text for this one
yes & M.\ Weik \newline 2000\vspace{-\baselineskip} & The \textit{Computer Science and Communications Dictionary} states \cite[p.\ 1326]{Weik2000}:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``\textbf{prime number:} A whole number that has no whole number divisors except 1 and itself, i.e., that when divided by a whole number other than 1 and itself will always produce a mixed number, i.e., a whole number and a fraction. \textit{Note:} The first few prime numbers are 1, 2, 3, 5, 7, 11, 13, [\textit{sic}] 19, and 23. Even numbers, except 2, products of two or more whole numbers, 0, mixed numbers, and repeating numbers such as 7777 \ldots or 3333 \ldots, are not prime numbers.'' \\
& & [Isn't 11 a repeating number?] \xskip \\
yes & J.\ B.\ Andreasen et al. \newline 2010\vspace{-\baselineskip} & The CliffsNotes preparation guide \cite[p.~342]{ASO2011} for an Elementary Education (K--6) teacher certification
test gives the following definition: % I have an image of this page stored
\vspace{-0.5\baselineskip} % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``\textbf{prime number:} A number with exactly two whole
number factors (1 and the number itself). The first few prime numbers are 1, 2, 3, 5, 7, 11, 13, and 17.'' \\
& & [Hopefully this is a typographical error as unity does not have ``\textit{exactly two} whole number factors.''] \xskip \\
% Contacted this source, offered to correct the entry, no response (June 2012)
yes & Carnegie Library of Pittsburgh \newline 2011\vspace{-2\baselineskip} & The \textit{Handy Science Answer Book} \cite[p.~13]{Handy2011} states:
\yskip % This give us the same gap that \\ would, but keeps the above from separating from what follows (do not remove the blank line below)
``A prime number is one that is evenly divisible
only by itself and 1. The integers 1, 2, 3, 5, 7, 11, 13, 17, and 19 are prime numbers. [\,\ldots] the largest known (and fortieth) prime number [\textit{sic}]: $2^{20996011}-1$. [\,\ldots] Mersenne primes occur where $2^{n-1}$ [\textit{sic}] is prime.'' \xskip \\
\end{longtable}
%\end{table}`
\end{center}
\section{Acknowledgements}
We would like to thank Professor David Broadhurst of Open University, UK, for help in finding some of these sources.
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11A41; Secondary 11A51, 01A55.
\noindent \emph{Keywords: } prime number, unity, history of mathematics.
\bigskip
\hrule
\bigskip
(Concerned with sequences
\seqnum{A000027},
\seqnum{A000040}, and
\seqnum{A008578}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received September 27 2012;
revised version received November 18 2012.
Published in {\it Journal of Integer Sequences}, December 27 2012.
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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