Journal of Integer Sequences, Vol. 15 (2012), Article 12.7.7

Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers

Vladimir Shevelev
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105

Gilberto García-Pulgarín
Universidad de Antioquia
Calle 67 No. 53-108
Medellín, Antioquia

Juan Miguel Velásquez-Soto
Departamento de Matemáticas
Universidad del Valle
Calle 13 No. 100-00
Cali, Valle del Cauca

John H. Castillo
Departamento de Matemáticas y Estadística
Universidad de Nariño
Calle 18 Carrera 50
San Juan de Pasto, Nariño


We introduce a new class of pseudoprimes, that we call "overpseudoprimes to base b", which is a subclass of the strong pseudoprimes to base b. Letting |b|n denote the multiplicative order of b modulo n, we show that a composite number n is an overpseudoprime if and only if |b|d is invariant for all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2p - 1, where p is prime, are overpseudoprimes to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well-known numbers are "primover to base b"; i.e., they are primes or overpseudoprimes to base b.

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(Concerned with sequences A002326 A141232 A141350 A141390 A178997.)

Received June 4 2012; revised version received September 6 2012. Published in Journal of Integer Sequences, September 8 2012.

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