Let \(k\geq 3\) be a positive integer; let \(a_1, \ldots, a_s\) be integers, and write \(F=\sum_{i=1}^s a_iX_i^k\). H. Davenport and D. J. Lewis, Proc. R. Soc. Lond., Ser, A 274, 443–460 (1963; Zbl 0118.28002)] have proved that if \(s\geq k^2 +1\) then \(F = 0\) has a non-trivial \(p\)-adic solution \(X\). The authors show that if is \(k\) odd and \(k > k_0(\varepsilon)\) then the weaker condition \(s > (2 + \varepsilon) k \log_2 k\) is sufficient; this is in fact an easier result than that of Davenport and Lewis, even though when applicable it is so much more precise. The authors further show by a counter-example that a condition \(s > k \log_2 k\) is necessary, and remark that standard techniques of Vinogradov (described by Davenport and Lewis) lead to the theorem that \(F = 0\) has a non-trivial rational solution so long as \(s > (4 + \varepsilon) k \log k\), \(k > k_1( \varepsilon)\).