2002: Preda Mihailescu proved Catalan’s conjecture – 8 and 9 are the only pair of consecutive perfect powers, thereby implying that no Wieferich prime pair actually solves the equation m^{p}-n^{q}=±1.

2000: Preda Mihailescu proved that any solutions p, q for m^{p}-n^{q}=±1 other than 2, 3 must be a Wieferich prime pair (=rare).

1976: For values of e^{245}< p, q there are no solutions for the equation m^{p}-n^{q}=±1.

1960: To solve m^{p}-n^{q}=1, p must divide n and q must divide m.

1951: For any pair m, n there is at most one solution for the equation m^{p}-n^{q}=1.

1844: Catalan’s conjecture – 8 and 9 are the only pair of consecutive perfect powers.

1738: Euler proved that if the exponents are 2,3, the only consecutive powers are 2^{3} and 3^{2}.

1320: Gersonides proved that if the bases are 2,3, the only consecutive powers are 2^{3} and 3^{2}.