寻找是强伪素数的Carmicheal数

Finding Carmicheal Numbers Which Are Also Strong Pseudoprimes

令N=q1q2q3,q1<q2<q3是三因子的Carmicheal数,定义C3,1-及C3,2-数,它们分别指qi=5 mod 8,i=1,2,3及qi≡5 mod 8,i=1,2,q3≡9 mod 16时的情况,它们有着较高的成为强伪素数的概率.本文首先给出成为这些数的充分必要条件然后给出算法,最后经过上机计算得到1024以内的有58个对于前5个素数基的C3,1-强伪素数,其中有一个是对于前8个素数基的强伪素数;以及27个对前4个素数基的C3,2-强伪素数,只有一个是对于前4个基的强伪素数.

Let N = q1q2q3, q1 〈 q2 〈 q3 be a 3 factors Carmicheal number, define C3,1 -and C3,2 -numbers, they respectively mean qi ≡ 5 mod 8, i = 1,2,3, and qi ≡ 5 mod 8, i = 1,2, q3 ≡ 9 mod 16, they also have a higher probability to be the strong pseudoprimes. In this paper we first give the sufficent and necessary condition of them and then the algorithems. Under the upper bound 10^24, we got 58 C3,1- spsp numbers to the first 4 primes bases, only 1 to the first 8; and 27 C3,2 -spsp numbers to the first 4 primes bases, only 1 to the first 5.