Evident formulae for a number of solutions of Diophantine equations and generalized solutions of Voroniy (Q2901601)

The present paper is devoted to the finding of solutions of the so-called Diophantine equations. More detail, given \(a_1,\ldots, a_n,\) \(k_1,\ldots, k_n,\) \(c\in {\mathbb N}\) it is proved that a number of integer solutions of the equations \(a_1x_1^{k_1}+\ldots+a_nx_n^{k_n}=c\) can be presented as a relation of some determinants connected with \(r=a_1c^{k_1}+\ldots+a_nc^{k_n},\) \(c\) and with some values of polynomial \(p(x)=\left(1+x^{a_1}+\ldots+x^{a_1c^{k_1}}\right)\) \(\ldots\left( 1+x^{a_n}+\ldots+x^{a_nc^{k_n}}\right)\). It is proved that equation \(x^{2k}+y^2+z^2=2mx^lyz\) has no any integer solution provided that \(k,l,m\in {\mathbb N}\) and \(x^2+y^2+z^2\neq 0\).