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1^p+2^p+...+n^p square

Source: France 1991 P1

May 14, 2021
number theory

Problem Statement

(a) Suppose that xn (n≥0)x_n~(n\ge0) is a sequence of real numbers with the property that x03+x13+…+xn3=(x0+x1+…+xn)2x_0^3+x_1^3+\ldots+x_n^3=(x_0+x_1+\ldots+x_n)^2 for each n∈Nn\in\mathbb N. Prove that for each n∈N0n\in\mathbb N_0 there exists m∈N0m\in\mathbb N_0 such that x0+x1+…+xn=m(m+1)2x_0+x_1+\ldots+x_n=\frac{m(m+1)}2. (b) For natural numbers nn and pp, we define Sn,p=1p+2p+…+npS_{n,p}=1^p+2^p+\ldots+n^p. Find all natural numbers pp such that Sn,pS_{n,p} is a perfect square for each n∈Nn\in\mathbb N.
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