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 (Acta Arith. 21, 1972, 251—259) proved the following result If a > 6 is a given integer, then for all large x * {n < x\n is pseudoprime and n = l(mod a)} > log x/ (2 log 2)a. We improved this result showing the following Theorem 2.13. ([10]) Let U = U(L,M) be a non degenerate Lehmer sequence and let a>\ be an integer with condition ((a,M) = 1. Then there is a positive constant C

6 such that for all large x, the number of all Lehmer pseudoprimes with parameters L, M which are congruent to 1 modulo a and not exceed x is greater than exp{(logx)

C

6}. For super Lehmer pseudoprimes we obtained the following Theorem 2.14. ([15]) Let U = U(L,M) be a non degenerate Lehmer sequence and let A denote the squarefree kernel of M. max(L,K), where K-L-4M. If A - +1 (mod 4), then for all large x the number of all super Lehmer pseudoprimes with parameters L, M not exceeding x is greater than (4Alog|a|)~Mogx, where a, ß denote the roots of z

2 - L

nz + M= 0 and | a\ >\ß\

. Showing a conjecture of A. Rotkiewicz, A. Makowski (Elem. Math. 29, 1974, 13) proved that the series II/logn, where n runs through all pseudoprimes to bace c, is 139