Catalan's conjecture

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archive posted 3 Aug 2026

The numbers $8 = 2^{3}$ and $9 = 3^{2}$ are consecutive perfect powers. Prove that they are the only such pair, that is, the only solution of $x^{a} - y^{b} = 1$ in integers $x, y, a, b > 1$ is $3^{2} - 2^{3} = 1$ .

Posed by Eugène Catalan, 1844

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diophantine-equations number-theory

History and references

Solved in 2002 by Preda Mihăilescu.

Catalan stated the conjecture in a note published in 1844, though special cases are much older. Levi ben Gershon (Gersonides) proved in 1343 the case where the bases are 2 and 3. In 1976 Robert Tijdeman used Baker's methods from transcendence theory to show that there are only finitely many solutions, with an explicit but astronomically large bound. Preda Mihăilescu finished the problem in April 2002 with an argument based on cyclotomic fields, and the proof was published in 2004.

References

Catalan's conjecture (Wikipedia)

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