Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation for exponents n and m greater than one, is finite.
[1][2] The theorem was proven by Dutch number theorist Robert Tijdeman in 1976,[3] making use of Baker's method in transcendental number theory to give an effective upper bound for x,y,m,n.
[1][4][5] Tijdeman's theorem provided a strong impetus towards the eventual proof of Catalan's conjecture by Preda Mihăilescu.
[6] Mihăilescu's theorem states that there is only one member of the set of consecutive power pairs, namely 9=8+1.
[7] That the powers are consecutive is essential to Tijdeman's proof; if we replace the difference of 1 by any other difference k and ask for the number of solutions of with n and m greater than one we have an unsolved problem,[8] called the generalized Tijdeman problem.