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

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 of 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

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 $Ay^{m}=Bx^{n}+k$ 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]

References

^ ^a ^b 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
^ 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
^ 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
^ Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN 978-0-387-90432-0, Zbl 0456.10006
^ 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
^ 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
^ 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
^ 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.
^ Narkiewicz (2011), pp. 253–254

[rnt20c-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

[ST202-8] 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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