Tijdeman's theorem

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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


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. Michel Langevin computed a value of exp exp exp exp 730 for the bound. [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 to the set of consecutive power pairs, namely 9=8+1. [7]

Generalized Tijdeman problem

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. It is conjectured that this set also will be finite. This would follow from a yet stronger conjecture of Subbayya Sivasankaranarayana Pillai (1931), see Catalan's conjecture, stating that the equation only has a finite number of solutions. The truth of Pillai's conjecture, in turn, would follow from the truth of the abc conjecture. [9]

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  6. Metsänkylä, Tauno (2004), "Catalan's conjecture: another old Diophantine problem solved" (PDF), Bulletin of the American Mathematical Society , 41 (1): 43–57, doi: 10.1090/S0273-0979-03-00993-5
  7. Mihăilescu, Preda (2004), "Primary Cyclotomic Units and a Proof of Catalan's Conjecture", Journal für die reine und angewandte Mathematik , 2004 (572): 167–195, doi:10.1515/crll.2004.048, MR   2076124
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