Catalan's conjecture

Solved, 2002Arithmetic and number theory

Statement

The only solution in integers $x, y, a, b$ with $a, b > 1$ of the equation $x^a - y^b = 1$ is $x=3, a=2, y=2, b=3$ — that is, $3^2 - 2^3 = 1$ , so 8 and 9 are the only two consecutive perfect powers of positive integers.

Eugène Charles Catalan posed the conjecture in an 1844 letter to Crelle's Journal, after noticing 8 and 9 seemed to be the only consecutive perfect powers among small numbers. Progress was slow: in 1976 Robert Tijdeman used Baker's theory of linear forms in logarithms to show that, if any solution besides $3^2-2^3=1$ exists, both exponents $a$ and $b$ must be bounded by an explicit (astronomically large) constant, reducing the problem to a finite — but computationally hopeless — search. Preda Mihăilescu found the final, purely algebraic argument in 2002, using properties of cyclotomic fields and Wieferich pairs to prove no other solution exists, without relying on Tijdeman's huge numerical bound; the result is now called Mihăilescu's theorem.

Solved by, 2002Preda Mihăilescu

Catalan's conjecture is the special case $k=1$ of a broader question about perfect powers: Pillai's conjecture (1931) asks whether, for every fixed positive integer $k$ , only finitely many pairs of perfect powers differ by exactly $k$ ; this remains open in general even though the $k=1$ case is now a theorem. The equation also generalizes to the Fermat–Catalan conjecture, which combines Fermat's Last Theorem and Catalan's conjecture by asking about integer solutions of $x^a+y^b=z^c$ with $1/a+1/b+1/c<1$ ; it remains open and is connected to the abc conjecture.

References

Preda Mihăilescu (2004). Primary cyclotomic units and a proof of Catalan's conjecture · DOI:10.1515/crll.2004.048
Robert Tijdeman (1976). On the equation of Catalan
René Schoof (2008). Catalan's Conjecture

History of the problem

1844Eugène Catalan proposes, in a letter published in Crelle's Journal, that 8 and 9 are the only consecutive perfect powers of positive integers greater than 1.
1850Victor-Amédée Lebesgue proves the special case with one exponent equal to 2 (there is no solution of $x^p - y^2 = 1$ for an odd prime $p$ other than the known one), settling an early piece of the conjecture by elementary methods.
1976Robert Tijdeman uses Alan Baker's theory of linear forms in logarithms to show any further solution of Catalan's equation must have both exponents bounded by an explicit, effectively computable constant, reducing the conjecture to a finite (though astronomically large) computation.
2000Preda Mihăilescu proves that any further solution would force $a$ and $b$ to form a 'double Wieferich pair', a very restrictive arithmetic condition satisfied by no known pair of exponents, sharply narrowing the search.Preda Mihăilescu
2002Preda Mihăilescu completes a fully algebraic proof of the conjecture, using the arithmetic of cyclotomic fields with no reliance on computer search or Tijdeman's numerical bound; the result becomes known as Mihăilescu's theorem.Preda Mihăilescu