Catalan’s Conjecture was proposed in 1844 by the Belgian mathematician Eugène Charles Catalan, who observed that although perfect powers such as \(2^3\), \(3^2\), and \(5^2\) appear frequently in number theory, no two of them seem to be consecutive integers. Catalan conjectured that the only exception was the pair \(8 = 2^3\) and \(9 = 3^2\). For more than 150 years, the problem resisted proof despite the efforts of many prominent mathematicians. This long‑standing puzzle was finally resolved in 2002, when Romanian mathematician Preda Mihăilescu announced a complete proof, now known as Mihăilescu’s Theorem.

Catalan’s Conjecture was proposed in 1844 by Eugène Charles Catalan. It claimed that the only pair of consecutive natural numbers that are both perfect powers is:

• \(8 = 2^3\)
• \(9 = 3^2\)

In 2002, Romanian mathematician Preda Mihăilescu proved the conjecture, ending 158 years of uncertainty. The result is now known as Mihăilescu’s Theorem.

Catalan’s Conjecture states that the only solution in natural numbers to

\[ x^a - y^b = 1\]

with \(x, y > 1\) and \(a, b > 1\) is

\[ 3^2 - 2^3 = 1.\]

In plain language:
8 and 9 are the only consecutive perfect powers in all of mathematics.

• It connects to deep algebraic number theory
• It resisted proof for more than a century
• It is a classic example of a simple problem with a very difficult proof
• It is now a foundational result in the study of Diophantine equations

• Born 1955 in Romania
• Worked at the University of Paderborn, Germany
• Announced the proof in 2002
• Full proof published in 2004
• His work uses cyclotomic fields and Galois module theory

• Perfect powers
• Diophantine equations
• Tijdeman’s theorem
• Pillai’s conjecture
• Catalan numbers (unrelated but often confused)