A variant of Waring’s problem

Abdullah N. Arslan
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 3, Pages 22–26
Full paper (PDF, 155 Kb)

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Abdullah N. Arslan
Department of Computer Science, Texas A&M University
Commerce, TX 75428, USA

Abstract

We introduce a variant of Waring’s problem. For a given positive integer k, consider the problem of writing any given positive integer N as the sum of the kth powers of consecutive integers starting at 1 using each of these kth powers (summands) exactly once, and repeating some of these summands as necessary. Let Ck denote the total number of such repeats. Determine minimum Ck for positive integers k required to write all positive integers using the kth powers of consecutive integers as described. We show that Ck ≤ g(k), where g(k) is the usual notation in Waring’s problem, the least number of non-negative kth powers sufficient to represent all positive integers. This result implies that for any given positive integer k, every positive integer N can be expressed as a linear combination of the kth powers of consecutive integers with positive integer coefficients that satisfy certain inequalities. Another implication is that for all positive integers N; n′; and k, the equation N = Σ_{i = 1}^n′i^kx_i has at least one solution (x₁, x₂, …, x_n′) in nonnegative integers if Σ_{i = 1}^n′i^k ≥ N.

Keywords

• Partitions
• Lagrange’s four square theorem
• Hilbert–Waring theorem
• function g

AMS Classification

• 11P05
• 11Y99
• 11D85
• 11B83

References

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