For any fixed integer \(k\geq2\), write here \({\mathcal E}_k\) for the set of positive integers that cannot be represented as the sum of two squares and at most two powers of \(k\), that is, as \(M^2+N^2+k^a+k^b\) nor \(M^2+N^2+k^a\) nor \(M^2+N^2\), where \(M\), \(N\), \(a\) and \(b\) are integers with \(a\), \(b\geq1\). The primary purpose of this paper is to prove that \({\mathcal E}_k\) is an infinite set for each \(k\geq 2\). In fact, the author presents a number of statements that are more detailed than the above one, and for example, the following is shown: \({\mathcal E}_k\) contains infinitely many even numbers, and when \(k\equiv 0, 3, 4, 6 \text{ or }7 \pmod 8\), the even numbers belonging to \({\mathcal E}_k\) have “positive density”, and moreover, unless \(k\equiv 3\pmod 4\), the same conclusions are obtained even if one relaxes the constraint on \(a\) and \(b\) in the above definition of \({\mathcal E}_k\) to “\(a\), \(b\geq 0\)”.
The proof is divided into cases, mainly according to \(k\) modulo 8, and some cases are fairly easy – for instance, when \(4\mid k\), every positive integer congruent to \(3 \pmod 4\) belongs to \({\mathcal E}_k\) obviously. Amongst others, the cases with \(k\equiv 2 \text{ and } 5 \pmod 8\) deserve particular attention. The most difficult, and perhaps the most interesting case is \(k=2\), and more than half of the paper is devoted to elaborate discussion on this case.