On the sum of two squares and two powers of \(k\). (English) Zbl 1173.11051
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.
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.
Reviewer: Koichi Kawada (Morioka)
MSC:
11P05 | Waring’s problem and variants |
Online Encyclopedia of Integer Sequences:
Numbers that cannot be expressed as the sum of two nonzero squares and at most two powers of two.Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.
Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.
Number of ways to write n as x^2 + y^2 + 2^z + 5*2^w, where x,y,z,w are nonnegative integers with x <= y.
Number of ways to write n as x^2 + y^k + 2^a + 2^b, where x,y,a,b are nonnegative integers with x >= y and a >= b, and k is either 2 or 3.