Catalan's conjecture

For all integers $x, y, a, b$ with $x > 0$ , $y > 0$ , $a > 1$ , $b > 1$ , the equation

x^{a} - y^{b} = 1

holds only for $x = 3$ , $a = 2$ , $y = 2$ , $b = 3$ ; that is, $3^{2} - 2^{3} = 1$ is the unique such representation, and $8$ and $9$ are the only two consecutive natural numbers that are both perfect powers with exponent greater than $1$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Catalan's conjecture on consecutive perfect powers
Let $x,y,p,q$ be integers with $x,y,p,q\geq 2$ . Catalan's conjecture. The equation
$x^p-y^q=1$
has only one solution, namely $(x,y,p,q)=(3,2,2,3)$ . This conjecture was stated in 1844 and was proved in 2002, so the asserted uniqueness is now a theorem.
source: Pranabesh Das and Amos Turchet, “Invitation to Integral and Rational points on curves and surfaces”, arXiv:1407.7750 (2015).

References

Primary source

Wikipedia

Additional references

Wikipedia, Catalan's conjecture, the article this problem comes from.

unsolved-problem

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Progress summary

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Claimed solved

The conjecture was proved in 2002: eight and nine are the only consecutive perfect powers with exponents greater than one.

Eugène Catalan proposed the conjecture in 1844. Preda Mihăilescu proved it in 2002, establishing the unique positive-integer solution $3^{2}-2^{3}=1$ .

Known results

Lebesgue (1850) and Ko Chao (1865) handled complementary cases that, together with Mihăilescu’s theorem, yield the full result.
Tijdeman (1976) proved that all solutions are bounded by an absolute effective constant.
Mihăilescu (2002; published 2004) proved that $x^{p}-y^{q}=1$ has no nonzero solutions for odd primes $p,q$ .

January 2026 proof exposition

Martin Klazar’s January 2026 arXiv preprint presents a self-contained exposition of Mihăilescu’s proof, including the odd-prime case and the cases with exponent $2$ . It reinforces the established theorem rather than proposing a new resolution.

Current status (as of August 2026): Catalan’s conjecture is completely settled; the only positive-integer solution is $(x,a,y,b)=(3,2,2,3)$ , and no part of the stated problem remains open.

Sources

No solutions have been posted yet.