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A is the square of a natural number

Source: Czech and Slovak MO, 2002, P3

September 15, 2012
number theorymodular arithmeticnumber theory unsolved

Problem Statement

Show that a given natural number AA is the square of a natural number if and only if for any natural number nn, at least one of the differences (A+1)2−A,(A+2)2−A,(A+3)2−A,⋯ ,(A+n)2−A(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A is divisible by nn.
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