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let d be a positive divisor of 5 + 1998^{1998}

Source: Vietnam TST 1998 for the 39th IMO, problem 5

June 26, 2005

modular arithmeticquadraticsnumber theory unsolvednumber theory

Problem Statement

Let

d

be a positive divisor of

5 + 1998^{1998}

. Prove that

d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2

, where

x, y

are integers if and only if

d

is congruent to 3 or 7

\pmod{20}