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 = xⁿ + 1,

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

The theorem was proven by Dutch number theorist Robert Tijdeman in 1976, and 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 to the set of consecutive power pairs, namely 9=8+1.

That the powers are consecutive is essential to Tijdeman's proof; if we replace a difference of one by any other difference k and ask for the number of solutions of

y^m = xⁿ + k

with n and m greater than one we have an unsolved problem. It is conjectured that this set also will be finite; its finiteness would follow, for instance, from the abc conjecture.

References

* Robert Tijdeman, On the equation of Catalan, Acta Arithmetica 29 (1976), pp. 197-209