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Problem 863

AIME late

Number theory Difficulty 5.7 Prove it

34. Let $p_{1}, p_{2}, \cdots, p_{n}$ be $n$ distinct prime numbers greater than 3, prove: $2^{p_{1} p_{2} \cdots p_{n}+1}$ has at least $4^{n}$ divisors.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

None

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None

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.