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In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation y^{m} = x^{n} + 1, for exponents n and m greater than one, is finite. The theorem was proven by Dutch number theorist Robert Tijdeman in 1976, and provided a strong impetus towards the eventual proof of Catalan's conjecture by Preda Mihăilescu. Mihăilescu's theorem states that there is only one member to the set of consecutive power pairs, namely 9=8+1. That the powers are consecutive is essential to Tijdeman's proof; if we replace a difference of one by any other difference k and ask for the number of solutions of y^{m} = x^{n} + k with n and m greater than one we have an unsolved problem. It is conjectured that this set also will be finite; its finiteness would follow, for instance, from the abc conjecture. References * Robert Tijdeman, On the equation of Catalan, Acta Arithmetica 29 (1976), pp. 197209 Retrieved from "http://en.wikipedia.org/"

