A new variant of the Hilbert-Waring problem

Abstract

In this paper, we propose a new variant of Waring's problem: to express a positive integer n as a sum of s positive integers whose product is a k-th power. We define, in a similar way to that done for g(k) and G(k) in Waring's problem, numbers g (k) and G (k). We obtain g (k) = 2k − 1,

Figurate Primes and Hilbert's 8th Problem

“…In [1], the first author raised a new variant of the Hilbert-Waring problem: to express a positive integer n as the sum of s positive integers whose product is a k-th power, i.e., n = x 1 + x 2 + · · · + x s such that…”

More Comprehensive Computational Number Theory

“…For example, recently, I found that except 2, 5 and 11, every positive prime can be expressed as a sum of three positive integers a, b, c, the product abc is a cube [1]. For instance, 3 = 1+1+1, 7 = 1+2+4, 13 = 1+3+9, 17 = 1+8+8, 19 = 4+6+9 and 1×1×1= 1 3 , 1×2×4=2 3 , 1×3×9=3 3 , 1×8×8=4 3 , 4×6×9=6 3 .…”

A new generalization of Fermat's Last Theorem

“…G ′ (k)) integers is a k-th power. We show [2] that g ′ (k) = 2k − 1; G ′ (p) ≤ p + 1; G ′ (2p) ≤ 2p + 2 (p ≥ 3); G ′ (4p) ≤ 4p + 2 (p ≥ 7); where k is a positive integer and p is prime. In this paper, we improve the results on composite numbers as follow.…”