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Infinitely many (x,y) such that x|P(y) and y|P(x)

Source: Romanian TST 1997

September 17, 2011

algebrapolynomialnumber theorygreatest common divisormodular arithmeticalgebra proposed

Problem Statement

Let

n\ge 2

be an integer and let

P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1

be a polynomial with positive integer coefficients. Suppose that

a_k=a_{n-k}

for all

k\in 1,2,\ldots,n-1

. Prove that there exist infinitely many pairs of positive integers

x,y

such that

x|P(y)

and

y|P(x)

.
Remus Nicoara