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prime divisors of n^{2}+n+1

Source: Ukrainian TST 2007 problem 12

September 19, 2007

polynomialDiophantine equationnumber theory proposednumber theory

Problem Statement

Prove that there are infinitely many positive integers

n

for which all the prime divisors of n^{2}\plus{}n\plus{}1 are not more then

\sqrt{n}

. Stronger one. Prove that there are infinitely many positive integers

n

for which all the prime divisors of n^{3}\minus{}1 are not more then

\sqrt{n}