Strong pseudoprimes to twelve prime bases

J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031–1037) some new kinds of pseudoprimes, called Sylow p p -pseudoprimes and elementary Abelian p p -pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow p p -pseudoprime to two bases only, where p = 2 p=2 or 3 3 . In this paper, in contrast to Browkin’s examples, we give facts and examples which are unfavorable for Browkin’s observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow 2 2 -pseudoprimes to the same bases. In Section 3, we give examples of Sylow p p -pseudoprimes to the first several prime bases for the first several primes p p . We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow p p -pseudoprime to the same bases for all p ∈ { 2 , 3 , 5 , 7 , 11 , 13 } p\in \{2,3,5,7,11,13\} . In Section 4, we define n n to be a k k - fold Carmichael Sylow pseudoprime , if it is a Sylow p p -pseudoprime to all bases prime to n n for all the first k k smallest odd prime factors p p of n − 1 n-1 . We find and tabulate all three 3 3 -fold Carmichael Sylow pseudoprimes > 10 16 >10^{16} . In Section 5, we define a positive odd composite n n to be a Sylow uniform pseudoprime to bases b 1 , … , b k b_1,\ldots ,b_k , or a Syl-upsp ( b 1 , … , b k ) (b_1,\ldots ,b_k) for short, if it is a Syl p _p -psp ( b 1 , … , b k ) (b_1,\ldots ,b_k) for all the first ω ( n − 1 ) − 1 \omega (n-1)-1 small prime factors p p of n − 1 n-1 , where ω ( n − 1 ) \omega (n-1) is the number of distinct prime factors of n − 1 n-1 . We find and tabulate all the 17 Syl-upsp ( 2 , 3 , 5 ) (2,3,5) ’s > 10 16 >10^{16} and some Syl-upsp ( 2 , 3 , 5 , 7 , 11 ) (2,3,5,7,11) ’s > 10 24 >10^{24} . Comparisons of effectiveness of Browkin’s observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6.

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.