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

TL;DR

This work addresses solvability and insolvability of high-degree Diophantine equations of the form $\sum_{i=1}^{m} x_i^{n}$ with nonnegative integers, for $n,m\ge 2$. It introduces the notion of a $\varphi$-divisor of $n$ and the canonical decomposition $d_{p_1\ldots p_l}$, proving that under suitable conditions the solution count remains unchanged when the right-hand side is multiplied by these canonical powers, i.e., $P_m^{n}(b(d_{p_1\ldots p_l})^{n})=P_m^{n}(b)$. Building on this reduction, the paper presents extensive insolvability criteria for $\sum_{i=1}^{m} x_i^{n}=b$ and extends them to standard forms $\sum_{i=1}^{m} x_i^{n}=b c^{n}$, giving concrete sufficient conditions (based on modular constraints, bounds, and $\varphi$-divisor structure) that guarantee nonexistence of natural solutions. The results include numerous explicit nonexistence cases, such as $x_1^{n}+x_2^{n}=(p^{s})^{n}$ for $n\ge 3$ and mixed right-hand sides $ (d_{p_1\ldots p_l})^{n}$, all derived via elementary number-theoretic arguments. Together, these criteria offer practical, verifiable insolvability tests and clarify when scaling the right-hand side preserves the solution count.

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 $\varphi(p_i^{k_i})\mid n$ $(p_i^{k_i}\geq 3)$.