Amdeberhan–Chen–Moll–Sagan valuation conjecture for generalized Fibonacci sequences

Let fn(s,t)f_n(s,t) denote the generalized Fibonacci sequence, and let ud u_d be the dd-adic valuation. For an integer dd', write deltadZ(n)=1delta_{d'\mathbb{Z}}(n)=1 if dd' divides nn and 00 otherwise. Amdeberhan–Chen–Moll–Sagan conjecture. Suppose s2s\ge 2 is an integer and d3d\ge 3 is an odd integer. There exist integers s(s,d)s'(s,d) and d(s,d)d'(s,d) such that ddd'\le d and

νd(fn(s,1))=δdZ(n)νd(snd).\nu_d(f_n(s,-1))=\delta_{d'\mathbb{Z}}(n)\nu_d\left(\frac{s'n}{d'}\right).

The paper states that the preceding valuation results prove this conjecture; the parser supplies no independent resolution evidence beyond that assertion.

Sources & referencesView supporting material

Primary source

Soohyun Park, “Arithmetic properties of generalized Fibonacci sequences”, arXiv:1407.8086 (2014).

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