A closed-form solution might be given by a tree. Valuations of quadratic polynomials

by A. Byrnes, J. Fink, G. Lavigne, I. Nogues, S. Rajasekaran, A. Yuan, L. Almodovar, X. Guan, A. Kesarwani, L. Medina, E. Rowland and V. Moll

Scientia, Series A: Math. Sciences 29 (2019) 11-28.
The \(p\)-adic valuation of an integer \(x\) is the largest power of the prime \(p\) that divides \(x\). It is denoted by \(\nu_p(x)\). This work describes properties of the valuation \(\nu_2(n^2+a)\), with \(a\in\mathbb{N}\). A distinction of the behavior of these valuations for \(a\equiv 7 \mod 8\) or not is presented.

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