Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

Bui Minh Phong: Recurrence sequences and pseudoprimes

A. Rotkiewicz (Math. Comp. 39, 1982, 239—247) imply that for the non-degenerate Lehmer sequence U(L,M) with L > 0 and K-L4M> 0 every arithmetic progression ax+b

, where (a,b

) = I, contains an infinite number of Euler-Lehmer pseudoprimes with parameters L and M. Using some theorems of A. Schinzel (Acta Arith. 8, 1963, 213—223) and J. Wójcik (Acta Arith. 40, 1982, 155—174; Acta Arith. 40,117—131) we proved the following Theorem 2.1. ([7]) LetU = U(L,M) be a non-degenerate Lehmer sequence and let s > 1 be an integer. Then there exists a positive integer w

0 such that for any integers a,b with condition (a,bw

o) = 1 and for infinitely many primes p of the form ax+b there exist an Euler-Lehmer pseudoprime which is the product of exactly s distinct primes and p is the least prime divisor of it Theorem 22. ([17]) LetU = U(L,M) be a non-degenerate Lehmer sequence with LK = L(L-4M) > Qand let a,s be positive integers. Then there are inßnitely many EulerLehmer pseudoprimes which are products of exactly s primes of the form ax

+1. A. Rotkiewicz (Bull. Acad. Polon. Sei. Ser. Sei. Math. Astr. Phys. 21, 1972, 793—797) showed that if R(A,B) is nondegenerate Lucas sequence for which B = 1 or B-1, and a,b are relatively prime integers, then there exist infinitely many composite numbers n of the form ax+b which satisfy 130