Overpseudoprimes, and mersenne and Fermat numbers as primover numbers

Vladimir Shevelev, Gilberto García-Pulgarín, Juan Miguel Velásquez-Soto, John H. Castillo

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
JournalJournal of Integer Sequences
Volume15
Issue number7
StatePublished - 8 Sep 2012

Keywords

  • Cyclotomic cosets of
  • Mersenne numbers
  • Modulo n
  • Order of
  • Overpseudoprime, Wieferich prime
  • Poulet pseudoprime
  • Super-poulet pseudoprime

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