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Is it true that \[\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}\] converges, where $p_n$ is the sequence of primes?
Erdős suggested that a computer could be used to explore this, and did not see any other method to attack this.

Tao [Ta23] has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.