×

On strong Lehmer pseudoprimes in the case of negative discriminant in arithmetic progressions. (English) Zbl 0822.11016

A Lehmer sequence \(P_ k\) with rational integer parameters \(L\) \((>0)\) and \(M\) is defined by \(P_ 0 =0\), \(P_ 1 =1\) and \(P_ n= LP_{n-1}- MP_{n-2}\) if \(n\) is odd and \(P_ n= P_{n-1}- MP_{n-2}\) if \(n\) is even. Its associated sequence \(V_ k\) is defined by the same recurrence relations with \(V_ 0=2\) and \(V_ 1=1\). For an odd composite integer \(n\) let \(n- (DL/n)= d\cdot 2^ s\), where \(D=L- 4M\), \((n,DL)=1\), \(d\) odd and \((DL/n)\) is the Jacobi symbol. If \(n\mid P_ d\) or \(n\mid V_{d2^ r}\) for some \(0\leq r<s\) then we say \(n\) is a strong Lehmer pseudoprime.
The author proves that for any nondegenerate sequence the arithmetic progression \(ax+b\) \((x= 0,1,2, \dots)\) with \((a,b)=1\) contain infinitely many odd strong Lehmer pseudoprimes. This result in the case \(D>0\) was proved by the author earlier [Math. Comput. 39, 239-247 (1982; Zbl 0492.10002)].
Reviewer: P.Kiss (Eger)

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A15 Power residues, reciprocity

Citations:

Zbl 0492.10002
PDFBibTeX XMLCite
Full Text: DOI EuDML