MathDB

20

Part of 1987 IMO Shortlist

Problems(1)

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

Source:

8/19/2010
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.
number theorypolynomialprime numbersquadraticsIMO ShortlistIMOIMO 1987