Fast primality tests for numbers less than 50⋅10⁹

Abstract

Consider the doubly infinite set of sequences A ( n ) A(n) given by \[ A ( n + 3 ) = r A ( n + 2 ) − s A ( n + 1 ) + A ( n ) A(n + 3) = rA(n + 2) - sA(n + 1) + A(n) \] with A ( − 1 ) = s A( - 1) = s , A ( 0 ) = 3 A(0) = 3 , A ( 1 ) = r A(1) = r . For a given pair r,s, the "signature" of n is defined to be the sextet \[ A ( − n − 1 ) , A ( − n ) , A ( − n + 1 ) , A ( n − 1 ) , A ( n ) , A ( n + 1 ) , A( - n - 1),A( - n),A( - n + 1),A(n - 1),A(n),A(n + 1), \] each reduced modulo n. Primes have only three types of signatures, depending on how they split in the cubic field generated by x 3 − r x 2 + s x − 1 {x^3} - r{x^2} + sx - 1 . An "acceptable" composite is a composite integer which has the same type of signature as a prime; such integers are very rare. In this paper, a description is given of the results of a computer search for all acceptable composites ⩽ 50 ⋅ 10 9 \leqslant 50 \cdot {10^9} in the Perrin sequence ( r = 0 , s = − 1 ) (r = 0,s = - 1) . Also, some numbers which are acceptable composites for both the Perrin sequence and the sequence with r = 1 r = 1 , s = 0 s = 0 are presented.