Finding 𝐶₃-strong pseudoprimes

Abstract

Let q 1 > q 2 > q 3 q_1>q_2>q_3 be odd primes and N = q 1 q 2 q 3 N=q_1q_2q_3 . Put \[ d = gcd ( q 1 − 1 , q 2 − 1 , q 3 − 1 ) and h i = q i − 1 d , i = 1 , 2 , 3. d=\gcd (q_1-1,q_2-1,q_3-1) \text { and } h_i=\tfrac {q_i-1}d, \;i=1,2,3. \] Then we call d d the kernel, the triple ( h 1 , h 2 , h 3 ) (h_1,h_2,h_3) the signature, and H = h 1 h 2 h 3 H=h_1h_2h_3 the height of N N , respectively. We call N N a C 3 C_3 -number if it is a Carmichael number with each prime factor q i ≡ 3 mod 4 q_i\equiv 3\mod 4 . If N N is a C 3 C_3 -number and a strong pseudoprime to the t t bases b i b_i for 1 ≤ i ≤ t 1\leq i\leq t , we call N N a C 3 C_3 -spsp ( b 1 , b 2 , … , b t ) (b_1, b_2,\dots ,b_t) . Since C 3 C_3 -numbers have probability of error 1 / 4 1/4 (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of ψ m \psi _m (the smallest strong pseudoprime to all the first m m prime bases). If we know the exact value of ψ m \psi _m , we will have, for integers n > ψ m n>\psi _m , a deterministic efficient primality testing algorithm which is easy to implement. In this paper, we first describe an algorithm for finding C 3 C_3 -spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 21978 C 3 C_3 -spsp ( 2 ) (2) ’s > 10 24 >10^{24} with heights > 10 9 >10^9 . We then give an overview of the 21978 C 3 C_3 - spsp(2)’s and tabulate 54 54 of them, which are C 3 C_3 -spsp’s to the first 8 8 prime bases up to 19 19 ; three numbers are spsp’s to the first 11 prime bases up to 31. No C 3 C_3 -spsp’s > 10 24 >10^{24} to the first 12 12 prime bases with heights > 10 9 >10^9 were found. We conjecture that there exist no C 3 C_3 -spsp’s > 10 24 >10^{24} to the first 12 12 prime bases with heights ≥ 10 9 \geq 10^9 and so that ψ 12 a m p ; = 3186 65857 83403 11511 67461 (24 digits) a m p ; = 399165290221 ⋅ 798330580441 , \begin{equation*} \begin {split} \psi _{12}&= 3186\; 65857\; 83403\; 11511\; 67461\; \text {(24 digits)} \\ &=399165290221\cdot 798330580441, \end{split}\end{equation*} which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those 21978 21978 C 3 C_3 -spsp ( 2 ) (2) ’s is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates N = q 1 q 2 q 3 N=q_1q_2q_3 of C 3 C_3 -spsp ( 2 ) (2) ’s and their prime factors q 1 , q 2 , q 3 q_1,q_2,q_3 to Miller’s tests, and obtain the desired numbers. At last we speed our algorithm for finding larger C 3 C_3 -spsp’s, say up to 10 50 10^{50} , with a given signature to more prime bases. Comparisons of effectiveness with Arnault’s and our previous methods for finding C 3 C_3 -strong pseudoprimes to the first several prime bases are given.