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Sum of consecutive squares

Source: American Mathematical Monthly - Romanian TST 2011

May 31, 2011
number theory proposednumber theory

Problem Statement

Show that: a) There are infinitely many positive integers nn such that there exists a square equal to the sum of the squares of nn consecutive positive integers (for instance, 22 is one such number as 52=32+425^2=3^2+4^2). b) If nn is a positive integer which is not a perfect square, and if xx is an integer number such that x2+(x+1)2+...+(x+n−1)2x^2+(x+1)^2+...+(x+n-1)^2 is a perfect square, then there are infinitely many positive integers yy such that y2+(y+1)2+...+(y+n−1)2y^2+(y+1)^2+...+(y+n-1)^2 is a perfect square.
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