A cyclotomic generalization of the sequence gcd(a(n)-1, b(n)-1)

作者：Cohen Joseph^*; Sonn Jack

来源：Journal de Theorie des Nombres de Bordeaux, 2015, 27(1): 53-65.

DOI：10.5802/jtnb.893

There has been interest during the last decade in properties of the sequence gcd(a(n) - 1, b(n) - 1), n = 1, 2, 3, ... , where a, b are fixed (multiplicatively independent) elements in one of Z, C[T], or F-q [T]. In the case of Z, Bugeaud, Corvaja and Zannier have obtained an upper bound exp(epsilon n) for any given epsilon > 0 and all large n, and demonstrate its sharpness by extracting from a paper of Adleman, Pomerance, and Rumely a lower bound exp (exp (c log n/log log n)) for infinitely many n, where c is an absolute constant. Silverman has proved an analogous lower bound deg gcd(a(n) - 1, b(n) - 1) >= cn for infinitely many n, over F-q [T]. This paper generalizes Silverman's theorem to gcd(Phi(m)(a(n)), Phi(m)(b(n))) for any positive integer m, where Phi(m)(x) is the mth cyclotomic polynomial, Silverman's result being the case m = 1. Over Z, the lower bound has been proved in the first author's Ph.D. thesis for the case m = 2, i.e. for gcd(a(n) + 1, b(n) + 1). Here we prove a conditional result that the lower bound for arbitrary m holds over Z under GRH (the generalized Riemann Hypothesis).

• 出版日期2015