Tijdeman's theorem
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
- [math]\displaystyle{ y^m = x^n + 1, }[/math]
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
- [math]\displaystyle{ y^m = x^n + k }[/math]
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 [math]\displaystyle{ A y^m = B x^n + k }[/math] 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]
References
- ↑ 1.0 1.1 Narkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Springer Monographs in Mathematics, Springer-Verlag, pp. 352, ISBN 978-0-857-29531-6
- ↑ Schmidt, Wolfgang M. (1996), Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, 1467 (2nd ed.), Springer-Verlag, p. 207, ISBN 978-3-540-54058-8
- ↑ Tijdeman, Robert (1976), "On the equation of Catalan", Acta Arithmetica 29 (2): 197–209, doi:10.4064/aa-29-2-197-209
- ↑ Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN 978-0-387-90432-0
- ↑ "Quelques applications de nouveaux résultats de Van der Poorten", Séminaire Delange-Pisot-Poitou, 17e Année (1975/76), Théorie des Nombres 2 (G12), 1977
- ↑ Metsänkylä, Tauno (2004), "Catalan's conjecture: another old Diophantine problem solved", Bulletin of the American Mathematical Society 41 (1): 43–57, doi:10.1090/S0273-0979-03-00993-5, http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf
- ↑ "Primary Cyclotomic Units and a Proof of Catalan's Conjecture", Journal für die reine und angewandte Mathematik 2004 (572): 167–195, 2004, doi:10.1515/crll.2004.048
- ↑ 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.
- ↑ (Narkiewicz 2011), pp. 253–254
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