Prove this equation has no integer solutions: $x^p_{1}+x^p_{2}+\cdots+x^p_{n}+1=(x_{1}+x_{2}+\cdots+x_{n})^2$

Question

Let $ p\equiv2\pmod 3$ be a prime number. Prove that the equation $x^p_{1}+x^p_{2}+\cdots+x^p_{n}+1=(x_{1}+x_{2}+\cdots+x_{n})^2$ has no integer solutions.

This problem is from the (Problems from the book) chapter 18 Quadratic reciprocity. Because this book problems have no answer. so How to use this methods to solve it?

Answer 1

Modulo $2$, the LHS is congruent to $x_1 + \cdots + x_n + 1$ while the RHS is congruent to $x_1 + \cdots + x_n$.