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Prime divisors of binary holonomic sequences. (English) Zbl 1165.11004

Suppose that \(\{u_n\}_{n=0}^\infty\) is a sequence of rational numbers satisfying the recurrence relation \[ f_0(n)u_{n+2}+f_1(n-1)u_{n+1}+f_2(n)u_{n}=0, \] for \(n\in{\mathbb N}\), where \(f_i(x)\in{\mathbb Q}[x]\) are not all zero for \(i=0,1,2\). Further suppose that \(\{u_n\}_{n=0}^\infty\) is not binary recurrent from some point on, i.e., there do not exist integers \(a\), \(b\) and \(c\) not all zero and \(n_0>0\) such that \[ au_{n+2}+bu_{n+1}+cu_{n}=0 \] holds identically for all \(n>n_0\). Let \[ U(N)=\prod_{^{\substack{ n\leq N\\ a_n\not=0}}}a_nb_n, \] where \(a_n\) is the numerator of \(u_n\) and \(b_n\) is the denominator of \(u_n\). In the present paper, the author proves that \(U(N)\) has at least \(c\log N\) distinct prime factors for all \(N>1\), where \(c\) is a positive constant depending on the sequence \(\{u_n\}_{n=0}^\infty\).
Reviewer: Hao Pan (Nanjing)

MSC:

11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
11L40 Estimates on character sums
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