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having at least 2d(n) prime factors

Source: Iran 3rd round 2011-number theory exam-p4

September 5, 2011
algebrapolynomialnumber theory proposednumber theory

Problem Statement

Suppose that nn is a natural number and nn is not divisible by 33. Prove that (n2n+nn+n+1)2n+(n2n+nn+n+1)n+1(n^{2n}+n^n+n+1)^{2n}+(n^{2n}+n^n+n+1)^n+1 has at least 2d(n)2d(n) distinct prime factors where d(n)d(n) is the number of positive divisors of nn. proposed by Mahyar Sefidgaran
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