TY - JOUR

T1 - Overpseudoprimes, and mersenne and Fermat numbers as primover numbers

AU - Shevelev, Vladimir

AU - García-Pulgarín, Gilberto

AU - Velásquez-Soto, Juan Miguel

AU - Castillo, John H.

PY - 2012/9/8

Y1 - 2012/9/8

N2 - 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 overpseu- doprime 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.

AB - 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 overpseu- doprime 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.

KW - Cyclotomic cosets of

KW - Mersenne numbers

KW - Modulo n

KW - Order of

KW - Overpseudoprime, Wieferich prime

KW - Poulet pseudoprime

KW - Super-poulet pseudoprime

UR - http://www.scopus.com/inward/record.url?scp=84880061900&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84880061900

VL - 15

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

IS - 7

ER -