# 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

## Contents

${\displaystyle y^{m}=x^{n}+1,}$

for exponents n and m greater than one, is finite. [1] [2]

## History

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

${\displaystyle y^{m}=x^{n}+k}$

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 ${\displaystyle Ay^{m}=Bx^{n}+k}$ 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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## References

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4. Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN   978-0-387-90432-0, Zbl   0456.10006 CS1 maint: discouraged parameter (link)
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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:
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
8. Shorey, Tarlok N.; Tijdeman, Robert (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. 87. Cambridge University Press. p. 202. ISBN   978-0-521-26826-4. MR   0891406. Zbl   0606.10011.
9. Narkiewicz (2011), pp. 253–254