AoPS - Problem

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Problem

Source: Irmo 2015 p2 q9

Tags: algebra, polynomial, integer root, Integer Polynomial

parmenides51

16.09.2018 11:47

Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.

Kaskade

16.09.2018 12:07

$2015=5\cdot13\cdot35$ and so there are

$8$ positive divisors of

$2015$ and

$16$ divisors including negative numbers. Clearly for each root both

$p\left(x\right),\ q\left(x\right)$ must take the value of a divisor of

$2015$ . As there are at least

$33$ such roots, by the pigeon hole principle both

$p\left(x\right),\ q\left(x\right)$ must take some value at least

$3$ times. It is well known that neither a non constant linear function nor quadratic can take the same value

$3$ times, and thus both

$p\left(x\right),\ q\left(x\right)$ must have degrees of at least

$3$