More exact values of the function \(\Gamma^\ast (k)\)
Let \(F(\mathbf {x})= a_1x_1^{k}+ a_2x_2^{k}+ \ldots +a_sx_s^{k}\) be a diagonal form with integer coefficients. Let \(\Gamma^{*}(k)\) be the smallest number \(s\) of variables needed to guarantee the existence of a nontrivial \(p\)-adic zero of \(F(\mathbf{x})\). [\textit{H. Davenport} and \textit{D. J. Lewis}, Proc. R. Soc. Lond. Ser. A 274, 443--460 (1963; Zbl 0118.28002)] proved that \(\Gamma^{*}(k)\leq k^{2}+1\), with equality when \(k+1\) is prime. The present paper provides the value of \(\Gamma^{*}(k)\) for all \(33 \leq k \leq 64\) for which this value is not already known. The method used is largely the same as the method in [\textit{M. Knapp}, J. Number Theory 131, No. 10, 1901--1911 (2011; Zbl 1241.11040)] which gave the value of \(\Gamma^{*}(k)\) for all \(k \leq 31\) for which this value was not already known.
scientific article; zbMATH DE number 7452409