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Tijdeman's theorem

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There are at most a finite number of consecutive powers

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

for exponents n and m greater than one, is finite.

History

The theorem was proven by Dutch number theorist Robert Tijdeman in 1976, 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.

Tijdeman's theorem 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 of the set of consecutive power pairs, namely 9=8+1.

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

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

with n and m greater than one we have an unsolved problem, 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 A y m = B x n + k {\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.

References

  1. ^ Narkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Springer Monographs in Mathematics, Springer-Verlag, p. 352, ISBN 978-0-857-29531-6
  2. Schmidt, Wolfgang M. (1996), Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467 (2nd ed.), Springer-Verlag, p. 207, ISBN 978-3-540-54058-8, Zbl 0754.11020
  3. Tijdeman, Robert (1976), "On the equation of Catalan", Acta Arithmetica, 29 (2): 197–209, doi:10.4064/aa-29-2-197-209, Zbl 0286.10013
  4. Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN 978-0-387-90432-0, Zbl 0456.10006
  5. Langevin, Michel (1977), "Quelques applications de nouveaux résultats de Van der Poorten", Séminaire Delange-Pisot-Poitou, 17e Année (1975/76), Théorie des Nombres, 2 (G12), MR 0498426
  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
  8. Shorey, Tarlok N.; Tijdeman, Robert (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. Vol. 87. Cambridge University Press. p. 202. ISBN 978-0-521-26826-4. MR 0891406. Zbl 0606.10011.
  9. Narkiewicz (2011), pp. 253–254
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