Rotkiewicz, A.; Wasén, R. Lehmer’s numbers. (English) Zbl 0436.10007 Acta Arith. 36, 203-217 (1980). Let \(L\), \(M\) and \(K=L- 4M\) be integers with conditions \(L>0\), \(M\ne 0\), \(K\ne 0\) and \((L,M) =1\). Define a sequence \(V_n\), of integers by \(V_0 = 2\), \(V_1 = 1\) and by the recursions \(V_{n+2} = LV_{n+1} - MV_n\) for even \(n\) and \(V_{n+2} = V_{n+1} - MV_n\) for odd \(n\). It is called the associated sequence of Lehmer’s numbers. We say an integer \(n\) satisfies the Lucas condition for the trinomial \(f = z^2 - \sqrt{L}z + M\) if \(n\) is odd, \((n,KL) = 1\), and \(n\mid V{[n-(KL/n)]/2}\). The authors give a reformulation of H. Riesel’s theorem on primality [Math. Comput. 23, 869–875 (1969; Zbl 0186.07803)], showing \(N=h\cdot 2^n -1\), where \(h\) is odd and \(h<2^n\), is a prime if and only if \(N\) satisfies the Lucas condition for a certain trinomial. They prove that for certain composite numbers the Lucas condition is also satisfied: If \(K>0\), then every arithmetical progression \(ax+b\), where \((a,b)=1\), which contains prime numbers satisfying the Lucas condition contains also infinitely many composite numbers which satisfy the Lucas condition for any fixed trinomial. Reviewer: Péter Kiss (Eger) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 Document MSC: 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11A51 Factorization; primality 11B25 Arithmetic progressions Keywords:primality; Lehmer’s numbers; Lucas condition Citations:Zbl 0186.07803 PDFBibTeX XMLCite \textit{A. Rotkiewicz} and \textit{R. Wasén}, Acta Arith. 36, 203--217 (1980; Zbl 0436.10007) Full Text: DOI EuDML