2 problems

Stickelberger's conjecture. For most fields $\mathbb{Q}(\zeta)$, every ideal of $\mathcal{O}_{\mathbb{Q}(\zeta)}$ is within a short enough ideal multiple from a principal ideal.

Eisenträger–Hallgren–Kitaev–Song conjecture. The methods of Eisenträger, Hallgren, Kitaev and Song should ultimately yield a quantum polynomial-time algorithm for solving the PIP.