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Lehmer’s numbers. (English) Zbl 0436.10007

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)

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A51 Factorization; primality
11B25 Arithmetic progressions

Citations:

Zbl 0186.07803
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