2012/06/04 by Vladimir Shevelev, Shevelev, Vladimir, Gilberto García-Pulgarín +5
Mathematics · #11A07 (Secondary) #11A51 (Primary) 11A41 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A07 #msc:11A41 #msc:11A51
paper · pdf · doi:10.48550/arxiv.1206.0606
9 pages
arxiv created 2012/06/04 · arxiv updated 2012/06/05
We introduce a new class of pseudoprimes-so called "overpseudoprimes to base b", which is a subclass of strong pseudoprimes to base b. Denoting via |b|n the multiplicative order of b modulo n, we show that a composite n is 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 overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base b.