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Schinzel’s conjecture H and divisibility in Abelian linear recurring sequences. (English) Zbl 0715.11008

Schinzel’s conjecture H says: if \(f_ 1(x),...,f_ k(x)\) are irreducible polynomials with integral coefficients and positive leading coefficients such that the product \(f_ 1(x)... f_ k(x)\) has no constant factor greater than 1, then there exist infinitely many positive integer x for which \(f_ 1(x),...,f_ k(x)\) are primes. For sequences of integers satisfying homogeneous linear recurrence relations it is proved: Let \(\{a^ i_ n\}\), \(i=1,...,r\) be a finite system of linear recurring sequences. Suppose that the characteristic polynomials \(g_ i(x)\), \(i=1,...,r\) have only simple roots and the splitting fields of the polynomials \(g_ i(x)\) have Abelian Galois groups over the rational number field. Then conjecture H implies the existence of infinitely many composite n such that \(a^ i_{ns}\equiv a^ i_ s(mod n)\) for every nonnegative integer s and for \(i=1,...,r\).
Reviewer: P.Kiss

MSC:

11B37 Recurrences
11A07 Congruences; primitive roots; residue systems
11R09 Polynomials (irreducibility, etc.)
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