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

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

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 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

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

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]