Equal Sums of Like Powers with Minimum Number of Terms

2016/02/28 by Choudhry, Ajai
#11D25 #11D41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1602.08698

Abstract

This paper is concerned with the diophantine system, ∑_i=1^s₁ x_i^r=∑_i=1^s₂ y_i^r, r=1, 2, …, k, where s₁ and s₂ are integers such that the total number of terms on both sides, that is, s₁+s₂, is as small as possible. We define β(k) to be the minimum value of s₁+s₂ for which there exists a nontrivial solution of this diophantine system. We find nontrivial integer solutions of this diophantine system when k < 6, and thereby show that β(2) =4, β(3) = 6, 7 ≤ β(4) ≤ 8 and 8 ≤ β(5) ≤ 10.

Related

• Simultaneous equal sums of three powers
• A Diophantine System
• Power values of sums of certain products of consecutive integers and\n related results
• On the simultaneous Diophantine equations m.(x₁^k+....+x_t1^k)=n.(y₁^k+....+y_t2^k); k=1,3
• Some Diophantine equations and inequalities with primes