Competition problem — Number theory

Number theory Difficulty 4.7 AIME Prove it 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 $(\tau(N)+\tau(N+1)+1) \equiv 3 \pmod{4}$ .

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.

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