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Problem 1011

AMC 12 late, AIME early

Number theory Difficulty 4.9 Prove it Estonian Math Competitions · Estonia

Do there exist integers $x$ and $y$ such that:

a. $x^2 + (x + 1)^2 + (x + 2)^2 = y^2$ ?

b. $x^2 + (x + 1)^2 + (x + 2)^2 + (x + 3)^2 = y^2$ ?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Answer: (a) No; (b) No.

a. The integers $x$ , $x+1$ , $x+2$ are congruent to $0$ , $1$ , $2$ modulo $3$ in some order. The squares of these integers are congruent to $0$ , $1$ , $1$ modulo $3$ , respectively. Hence the l.h.s. of the equation is congruent to $2$ while the r.h.s. is congruent to $0$ or $1$ modulo $3$ . Thus the equality cannot hold.

b. Among the integers $x$ , $x+1$ , $x+2$ , $x+3$ , there are two even numbers and two odd numbers. The squares of even numbers are divisible by $4$ while the squares of odd numbers are congruent to $1$ modulo $4$ . Thus the l.h.s. of the equation is congruent to $2$ while the r.h.s. is congruent to $0$ or $1$ modulo $4$ . Hence the equality cannot hold.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.