IMO 1992 SL 16

Prove that N = 5125−1

imoshortlistmathematicsolympiad

IMO 1992 SL 16

Origin: KOR

Problem

Prove that N = 5125−1 525−1 is a composite number.

Solution

Observe that x4 + x3 + x2 + x + 1 = (x2 + 3x + 1)2 −5x(x + 1)2. Thus for x = 525 we have N = x4 + x3 + x2 + x + 1 = (x2 + 3x + 1 −513(x + 1))(x2 + 3x + 1 + 513(x + 1)) = A \cdot B. Clearly, both A and B are positive integers greater than 1.