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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 ym = xn + 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 Mih ilescu. Mih ilescu''s theorem states that there is only one member to the set of consecutive power pairs, namely 9=8+1.