Two kinds of strong pseudoprimes up to $10^{36}$

Abstract

Let n > 1 be an odd composite integer. Write n - 1 = 2 s d with d odd. If either b d ≡ 1 mod n or b 2r d ≡ -1 mod n for some r = 0,1,..., s - 1, then we say that n is a strong pseudoprime to base b, or spsp(b) for short. Define t to be the smallest strong pseudoprime to all the first t prime bases. If we know the exact value of t , we will have, for integers n 10 36 ; and to give reasons and numerical evidence of K2- and C 3 -spsp's 12, where ψ' t (resp. ψ t ) is the smallest K2- (resp. C 3 -) strong pseudoprime to all the first t prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's < 10 36 to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: n = pq with p, q primes and q - 1 = 2(p - 1); and that a C 3 -spsp is an spsp and a Carmichael number of the form: n = q 1 q 2 q 3 with each prime factor q i = 3 mod 4.).