The Hasse-pair coverage conjecture

A pair of prime powers $(q_1,q_2)$ is a Hasse pair when

$$\lvert\sqrt{q_1}-\sqrt{q_2}\rvert\leq 1.$$

If both prime powers are odd, it is an odd Hasse pair. Hasse-pair coverage conjecture. Every prime power is a member of some Hasse pair, and every odd prime power is a member of some odd Hasse pair. The conjecture is known to hold for all prime powers below $10^{19}$ and is intimately connected with Andrica's conjecture; its general validity remains open.