On the representation of integers by \(p\)-adic diagonal forms (Q1139620)

Let \(p\) be an odd prime, let \(d\) be a positive integer such that \((d,p-1)=1\), let \(r\) denote the \(p\)-adic valuation of \(d\) and let \(m=1+3+3^2+\ldots+3^r\). It is shown that for every \(p\)-adic integer \(n\) the equation \(\sum_{i=1}^m X_i^d=n\) has a nontrivial \(p\)-adic solution. It is also shown that for all \(p\)-adic units \(a_1, a_2, a_3, a_4\) and all \(p\)-adic integers \(n\) the equation \(\sum_{i=1}^4 a_iX_i^p=n\) has a nontrivial \(p\)-adic solution. A corollary to each of these results is that every \(p\)-adic integer is a sum of four \(p\)th powers of \(p\)-adic integers.