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 mp-nq=±1.
2000: Preda Mihailescu proved that any solutions p, q for mp-nq=±1 other than 2, 3 must be a Wieferich prime pair (=rare).
1976: For values of e245< p, q there are no solutions for the equation mp-nq=±1.
1960: To solve mp-nq=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 mp-nq=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 23 and 32.
1320: Gersonides proved that if the bases are 2,3, the only consecutive powers are 23 and 32.To the MNS presentation