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Problem 907

AMC 12 late, AIME early

Number theory Difficulty 4.7 Prove it Problems of Ukrainian Authors · Ukraine

Let $\tau(n)$ be the number of divisors of a natural number $n$ . Prove that there exists infinitely many natural numbers $N$ such that $(\tau(N)+\tau(N+1)+1) \equiv 3 \pmod{4}$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

So, finally, we have that every number $N = 3n^3$ with $n = 24m+1$ satisfies the problem condition, which evidently means, that there are infinitely many such numbers.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.