## On the sum of two squares and two powers of $k$

### Volume 112 / 2008

Colloquium Mathematicum 112 (2008), 235-267
MSC: Primary 11P32; Secondary 11A41.
DOI: 10.4064/cm112-2-3

#### Abstract

It can be shown that the positive integers representable as the sum of two squares and one power of $k$ ($k$ any fixed integer $\geq 2$) have positive density, from which it follows that those integers representable as the sum of two squares and (at most) two powers of $k$ also have positive density. The purpose of this paper is to show that there is an infinity of positive integers
*not* representable as the sum of two squares and two (or fewer) powers of $k$, $k$ again any fixed integer $\geq 2$.