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
for exponents n and m greater than one, is finite.[1][2]
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]
References
^ abNarkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Springer Monographs in Mathematics, Springer-Verlag, p. 352, ISBN978-0-857-29531-6
^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), MR0498426