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

Abstract

The sufficient conditions for insolvability of the Diophantine equation ${\sum }_{i=1}^m{x}_i^n=b{c}^n$ ($n,m\ge 2$, $b,c\in \mathbb{N}$) 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 $\phi (p_i^{k_i})\mid n$ ($p_i^{k_i}\ge 3)$ for some natural numbers $k_i$ $(i=1,2,\dots ,l)$; $\phi (x)$ is the Euler's totient function. Moreover, it is proved that if $b<m<p_i^{k_i}$ $(i=1,2,\dots ,l)$, then this equation has no solution with natural components $x_1,x_2,\dots ,x_m$. Besides, applying only elementary methods, it is proved that the Diophantine equation $x_1^n+x_2^n=(p^s{p}_1^{s_1}{p}_2^{s_2}\dots p_l^{s_l}{)}^n$ (with nonnegative integers $s$, $s_i$ $(i=1,2,..,l)$) has no solution with natural components if $n\ge 3$, $p$ is a prime number, while $p_i$ is a prime such that there is a natural number $k_i$ with $\phi (p_i^{k_i})\mid n$ $(p_i^{k_i}\ge 3)$.

19 pages; in Russian