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Sum of perfect squares

Source: Turkish TST 2012 Problem 5

March 26, 2012
number theory proposednumber theory

Problem Statement

A positive integer nn is called good if for all positive integers aa which can be written as a=n2∑i=1nxi2a=n^2 \sum_{i=1}^n {x_i}^2 where x1,x2,…,xnx_1, x_2, \ldots ,x_n are integers, it is possible to express aa as a=∑i=1nyi2a=\sum_{i=1}^n {y_i}^2 where y1,y2,…,yny_1, y_2, \ldots, y_n are integers with none of them is divisible by n.n. Find all good numbers.
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