×

On super Lucas and super Lehmer pseudoprimes. (English) Zbl 0597.10004

Let \(U_ n=U_ n(L,M)=(\alpha^ n-\beta^ n)/(\alpha -\beta)\) for odd n and \(U_ n=(\alpha^ n-\beta^ n)/(\alpha^ 2-\beta^ 2)\) for even n denote the n-th Lehmer number, where \(\alpha\),\(\beta\) are roots of \(f(z)=z^ 2-\sqrt{L} z+M\). Let \((L,M)=1\), LM\(\neq 0\), \(K=L-4M\neq 0\), and \(\alpha\) /\(\beta\) be not a root of unity. A composite number n is called a super Lehmer pseudoprime with parameters L,M if \((n,LMK)=1\) and each divisor d of it satisfies the congruence \(U_{d-(LK/d)}\equiv 0 (mod d),\) where (LK/d) is the Jacobi symbol.
The author proves that there exists a positive integer \(w_ 0\) such that for infinitely many primes p of the form \(ax+b\), where \((a,b)=1\) and \(b\equiv 1 (mod(a,w_ 0))\), there are primes q and r such that pqr is a super Lehmer pseudoprime with parameters L,M. If \(LK>0\), then for every integer \(a>1\) there are infinitely many triplets of distinct primes p,q and r of the form \(ax+1\) such that pqr is a super Lehmer pseudoprime with parameter L,M.

MSC:

11A15 Power residues, reciprocity
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B37 Recurrences
PDFBibTeX XMLCite