Conjectured number of solutions to p^i - q^j = k^2, where p=prime(n), q=prime(n+1), and k an integer.
5, 3, 1, 2, 1, 4, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1
1
Ventullo proves the numbers for the first three primes.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Alessandro Ventullo, Difference of powers of consecutive primes which are perfect squares, arXiv 1604.05334 (Apr 18 2016)
(Mma) p = 2; mxPwr = 100; Table[q = NextPrime[p]; cnt = 0; Do[If[IntegerQ[Sqrt[q^i - p^j]], cnt++], {i, 0, mxPwr}, {j, 0, i*Log[q]/Log[p]}]; p = q; cnt, {100}]
nonn
T. D. Noe, Apr 22 2016