The sufficient conditions for insolvability of some Diophantine equations of $n$-th degree

TL;DR

This paper establishes sufficient conditions for the insolvability of certain high-degree Diophantine equations involving sums of powers, using elementary methods and properties of prime factorization and Euler's totient function.

Contribution

It provides new criteria for the non-existence of solutions to specific Diophantine equations based on prime decomposition and Euler's totient divisibility conditions.

Findings

No solutions when b < m < p_i^{k_i} under given conditions.

Proves non-solvability of specific equations for n ≥ 3 with prime power factors.

Uses elementary methods to derive insolubility results without advanced techniques.

Abstract

The sufficient conditions for insolvability of the Diophantine equation $\sum_{i=1}^{m}x_i^{n}=bc^{n}$ ($n, m \geq 2$, $b, c\in \mathbb{N}$) in nonnegative integers are obtained for the case where the canonical decomposition of the number $c$ consists of powers of primes $p_i$ which satisfy the condition $\varphi(p_i^{k_i})\mid n$ ($p_i^{k_i }\geq 3)$ for some natural numbers $k_i$ $(i=1,2,\ldots ,l)$; $\varphi(x)$ is the Euler's totient function. Moreover, it is proved that if $b< m< p_i^{k_i}$ $(i=1,2,\ldots ,l)$, then this equation has no solution with natural components $x_1,x_2,\ldots ,x_m$. Besides, applying only elementary methods, it is proved that the Diophantine equation $x_1^n+x_2^n=(p^{s} p_1^{s_1} p_2^{s_2}\ldots p_l^{s_l})^{n}$ (with nonnegative integers $s$, $s_i$ $(i=1,2,..,l)$) has no solution with natural components if $n\geq 3$, $p$ is a prime number, while $p_i$ is a prime such that there is a natural number $k_i$ with $\varphi(p_i^{k_i})\mid n$ $(p_i^{k_i}\geq 3)$.