MathDB

Primes and values attained by a polynomial

Source: Iberoamerican Olympiad 1990, Problem 3

May 21, 2007

algebrapolynomialnumber theory proposednumber theory

Problem Statement

Let

b

,

c

be integer numbers, and define

f(x)=(x+b)^2-c

.
i) If

p

is a prime number such that

c

is divisible by

p

but not by

p^{2}

, show that for every integer

n

,

f(n)

is not divisible by

p^{2}

.
ii) Let

q \neq 2

be a prime divisor of

c

. If

q

divides

f(n)

for some integer

n

, show that for every integer

r

there exists an integer

n'

such that

f(n')

is divisible by

qr

.

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