Almodovar, Leyda; Byrnes, Alyssa N.; Fink, Julie; Guan, Xiao; Kesarwani, Aashita; Lavigne, Gary; Medina, Luis A.; Moll, Victor H.; Nogues, Isabelle; Rajasekaran, Senthil; Rowland, Eric; Yuan, Amber A closed-form solution might be given by a tree. Valuations of quadratic polynomials. (English) Zbl 1430.11004 Sci., Ser. A, Math. Sci. (N.S.) 29, 11-28 (2019). Summary: 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. Cited in 3 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11S05 Polynomials Keywords:valuations; quadratic polynomials; congruences modulo 8 PDFBibTeX XMLCite \textit{L. Almodovar} et al., Sci., Ser. A, Math. Sci. (N.S.) 29, 11--28 (2019; Zbl 1430.11004) Full Text: Link References: [1] F. Aragon, D. Bailey, J. Borwein, and P. Borwein. Walking on real numbers.Mathematical Intelligencer, 35:42-60, 2013. · Zbl 1282.11088 [2] J. M. Borwein and R. Crandall. Closed forms: what they are and why we care.Notices Amer. Math. Soc., 60:50-65, 2013. · Zbl 1334.33042 [3] T. Chow. What is a closed-form number?Amer. Math. Monthly, 106:440-448, 1999. · Zbl 1003.11063 [4] F. Gouvea.p-adic numbers. Springer-Verlag, 2nd edition, 1997. · Zbl 0874.11002 [5] G. H. Hardy, D. R. Wright, E. M.; revised by Heath-Brown, and J. Silverman.An Introduction to the Theory of Numbers. Oxford University Press, 6th edition, 2008. · Zbl 1159.11001 [6] N. Koblitz.p-adic Numbers,p-adic Analysis, and Zeta functions, volume 58 ofGraduate Texts in Mathematics. Springer Verlag, New York, 1984. · Zbl 0364.12015 [7] A. M. Legendre.Th´eorie des Nombres. Firmin Didot Fr‘eres, · JFM 60.1212.16 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.