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All numbers f(0),f(1),...,f(p-2) are primes(IMO SL 1987-P20)

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August 19, 2010
number theorypolynomialprime numbersquadraticsIMO ShortlistIMOIMO 1987

Problem Statement

Let n≥2n\ge2 be an integer. Prove that if k2+k+nk^2+k+n is prime for all integers kk such that 0≤k≤n30\le k\le\sqrt{n\over3}, then k2+k+nk^2+k+n is prime for all integers kk such that 0≤k≤n−20\le k\le n-2.(IMO Problem 6)
Original Formulation
Let f(x)=x2+x+pf(x) = x^2 + x + p, p∈N.p \in \mathbb N. Prove that if the numbers f(0), f(1), \cdots , f( \sqrt{p\over 3} ) are primes, then all the numbers f(0),f(1),⋯ ,f(p−2)f(0), f(1), \cdots , f(p - 2) are primes.
Proposed by Soviet Union.
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