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Catalan's conjecture is Mihăilescu's theorem

This text, derived from lecture notes for a winter 2025/26 course, aims to provide a complete exposition of Mihăilescu's proof of Catalan's conjecture by systematically developing the necessary number-theoretic results from Euler's and Lebesgue's theorems through Cassels' relations to the final theorem.

Original authors: Martin Klazar

Published 2026-01-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Martin Klazar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a very specific, stubborn mystery in the world of numbers. The case is called Catalan's Conjecture, and it was first proposed in 1844 by a mathematician named Eugène Catalan.

Here is the mystery in plain English:
In the world of whole numbers, we have "perfect powers." These are numbers like squares ( $2^2=4$ , $3^2=9$ ), cubes ( $2^3=8$ , $3^3=27$ ), or fourth powers ( $2^4=16$ ).
Catalan noticed that there is exactly one pair of perfect powers that sit right next to each other on the number line: 8 and 9.

$8 = 2^3$ (a cube)
$9 = 3^2$ (a square)
They are consecutive ( $9 - 8 = 1$ ).

Catalan guessed that this is the only time this happens. He claimed that if you look for any other two perfect powers that are neighbors (like $x^m - y^n = 1$ ), you will never find another pair.

For 160 years, this was just a guess. Then, in 2004, a mathematician named Preda Mihăilescu proved it was true. This paper, written by Martin Klazar, is a guidebook that walks you through Mihăilescu's proof, breaking it down into manageable steps.

Here is how the paper explains the solution, using simple analogies:

1. The Easy Cases (The "Low Hanging Fruit")

Before tackling the hardest part, the paper clears out the easy scenarios where one of the exponents is a small number (like 2 or 3). Think of this as checking the obvious suspects first.

The Square vs. Cube Case ( $x^2 - y^3 = 1$ ):
The paper looks at the equation where a square minus a cube equals 1. It uses a method similar to Euler's old trick. Imagine you are trying to fit a square peg into a round hole. The math shows that the only way this fits is if the numbers are 8 and 9 (or some trivial zeros). The author, Klazar, actually provides three different ways to solve this specific puzzle, like showing a lock can be opened with three different keys.
The High Power vs. Square Case ( $x^m - y^2 = 1$ ):
Here, we have a huge power (like $x^5$ ) minus a square equals 1. The paper uses a tool called Gaussian Integers (numbers with an imaginary part, like $a + bi$). Think of this as switching from a 2D map to a 3D map to see the problem from a new angle. It proves that no matter how high the power goes, you can't find a neighbor square.
The Square vs. High Power Case ( $x^2 - y^q = 1$ ):
This is the reverse: a square minus a huge power equals 1. A mathematician named Chao Ko proved this long ago. The paper explains his logic: if such a pair existed, the numbers would have to behave in a way that creates a mathematical contradiction, like a clock running backward.

2. The Hard Part (The "Mountain Peak")

Once the small numbers are ruled out, the paper tackles the real monster: Two distinct odd prime numbers (like 7 and 11, or 13 and 17) as exponents.
The equation is: $x^p - y^q = 1$ .

This is where Mihăilescu's genius comes in. The paper outlines his proof as a series of logical traps that force the numbers to admit they don't exist.

The "Cassels Relations" (The Divisibility Trap):
Imagine you have two suspects, $x$ and $y$ . The paper shows that if they exist, they must have very specific "family ties."
- One relation says: "If you exist, your number $x$ must be divisible by the exponent $q$ ."
- Another says: "Your number $y$ must be divisible by the exponent $p$ ."
  It's like saying, "If you are a spy, you must have a red hat and a blue shoe." The paper proves these conditions are necessary.
The "Super-Cassels Relations" (The Double Trap):
Mihăilescu went deeper. He proved that the divisibility isn't just once; it's squared.
- $x$ must be divisible by $q^2$ .
- $y$ must be divisible by $p^2$ .
  This is like saying, "Not only do you need a red hat, you need a giant red hat." This makes the numbers incredibly large and restrictive.
The "Obstruction Group" (The Wall):
The proof then moves into a very abstract area called Algebraic Number Theory. Imagine the numbers are trying to climb a mountain, but there is a magical wall (an "obstruction group") that stops them.
The paper explains that the structure of these numbers (specifically in "Cyclotomic Fields," which are like special coordinate systems for roots of unity) creates a barrier. The "Stickelberger ideal" is a fancy name for a rule that says, "You cannot climb this wall."

3. The Final Knockout (The "M4" Theorem)

The paper culminates in Theorem M4. This is the final blow.
Mihăilescu showed that if you combine all the previous rules (the divisibility by squares, the size of the numbers, and the properties of the "wall"), the only possible candidates for the exponents $p$ and $q$ are the tiny numbers 3 and 5.

But wait! The paper already proved in the earlier chapters that if the exponents are 3 or 5, the equation has no solution (except the trivial ones).

If the exponents are big (7, 11, etc.), the "wall" stops them.
If the exponents are small (3, 5), the earlier "easy case" proofs stop them.

The Conclusion:
There is no place left for a solution to hide. The only consecutive perfect powers in the entire universe of numbers are 8 and 9.

Summary

This paper is a complete, step-by-step manual on how to prove that 8 and 9 are the only neighbors in the family of perfect powers.

It starts with simple puzzles (squares and cubes).
It moves to complex tools (imaginary numbers and divisibility rules).
It builds a massive logical structure (the obstruction group) that proves no other pair can ever exist.

It's a story of how mathematicians used a combination of old tricks and brand-new, high-tech mathematical machinery to solve a 160-year-old riddle.

Technical Summary: Catalan's Conjecture is Mihăilescu's Theorem

Problem Statement
The paper addresses Catalan's conjecture, originally proposed by Eugène Ch. Catalan in 1844. The conjecture asserts that the only solution in natural numbers to the Diophantine equation $x^m - y^n = 1$ for $m, n \ge 2$ and $x, y > 0$ is the pair $(x, y, m, n) = (3, 2, 2, 3)$ , corresponding to $3^2 - 2^3 = 1$ . The text aims to provide a complete, self-contained proof of this conjecture, which was proven by Preda Mihăilescu in 2004. The proof is structured to handle the equation by reducing the exponents to distinct odd primes $p$ and $q$ , thereby dividing the problem into an "elementary" part (where one exponent is 2) and a "non-elementary" part (where both exponents are $\ge 3$ ).

Methodology and Structure
The text is organized as a comprehensive exposition of Mihăilescu's proof, supplemented by necessary background in algebraic number theory. The methodology proceeds in three distinct phases:

Elementary Cases (Chapters 1–3): The author resolves cases where at least one exponent is 2.
- Chapter 1 ( $x^2 - y^3 = 1$ ): Three distinct elementary resolutions are presented: a modern completion of Euler's 1737 argument using elliptic curves and rational points; a 1989 resolution by the author (Klazar) utilizing factorization and properties of Pell equations ( $x^2 - 3y^2 = 1$ ); and a resolution of the related equation $x^3 + y^3 = 2z^3$ by A. Wakulicz.
- Chapter 2 ( $x^m - y^2 = 1$ for odd $m \ge 3$ ): The text presents V. Lebesgue's 1850 proof, which relies on the unique factorization domain (UFD) property of the Gaussian integers $\mathbb{Z}[i]$ . The author proves that Euclidean domains are UFDs to establish this property rigorously.
- Chapter 3 ( $x^2 - y^q = 1$ for prime $q \ge 5$ ): The text details Chao Ko's 1965 proof, utilizing properties of cyclotomic fields and specific lemmas regarding the greatest common divisor of factors in the equation.
Divisibility Relations (Chapter 4): The text establishes two critical divisibility conditions for hypothetical non-trivial solutions to $x^p - y^q = 1$ (where $p > q > 2$ are distinct primes). Following John W. S. Cassels, the author proves that $q \mid x$ and $p \mid y$ . These relations are derived using elementary number theory and properties of $p$ -adic orders.
The Non-Elementary Case (Chapters 5–8): This section outlines Mihăilescu's proof for the case where both exponents are odd primes.
- Chapter 5: The author outlines the logical deduction of the main theorem from four specific theorems (M1–M4) due to Mihăilescu. These theorems establish:
  - M0.95/M1: Super-Cassels relations ( $p^2 \mid y$ and $q^2 \mid x$ ) and double Wieferich conditions ( $pq^{-1} \equiv 1 \pmod{q^2}$ and $qp^{-1} \equiv 1 \pmod{p^2}$ ).
  - M2: A congruence condition ( $p \equiv 1 \pmod q$ or $q \equiv 1 \pmod p$ ) for primes $p, q \ge 7$ .
  - M3: Bounds on the size of the primes ( $p < 4q^2$ and $q < 4p^2$ ).
  - M4: The exclusion of solutions where one prime is 3 or 5.
- Chapters 6–8: These chapters provide the necessary algebraic machinery for the non-elementary proofs. They introduce number fields, rings of integers, and cyclotomic fields. Specifically, Chapter 6 defines the "obstruction group" and establishes that the class numbers of $\mathbb{Q}(\zeta_3)$ and $\mathbb{Q}(\zeta_5)$ are 1 (implying their rings of integers are UFDs). Chapter 7 outlines the derivation of the super-Cassels relations using the Stickelberger ideal, and Chapter 8 addresses the specific cases involving the primes 3 and 5.

Key Contributions and Results

Self-Contained Proof: The primary contribution of this text is the presentation of a complete and self-contained proof of Catalan's conjecture. The author notes that existing books on the subject (e.g., by Ribenboim, Schoof, Bilu et al.) are not self-contained and rely on results taken for granted. This text fills those gaps, particularly in the elementary sections and the algebraic foundations required for Mihăilescu's work.
Multiple Resolutions: The text offers three distinct proofs for the case $x^2 - y^3 = 1$ , including the author's own 1989 elementary resolution and a modernized version of Euler's argument.
Rigorous Foundations: The text explicitly proves that Euclidean domains are UFDs and applies this to $\mathbb{Z}[i]$ and cyclotomic fields $\mathbb{Z}[\zeta_3]$ and $\mathbb{Z}[\zeta_5]$ , ensuring the necessary algebraic properties are established within the text rather than assumed.
Logical Synthesis: The paper synthesizes the elementary results (Chapters 1–3) with the advanced algebraic number theory (Chapters 6–8) to demonstrate that no solutions exist for distinct odd primes, thereby completing the proof of the conjecture.

Significance
The paper claims significance as a pedagogical and technical resource that demystifies Mihăilescu's proof. By providing a "complete and self-contained presentation," it aims to make the proof accessible to readers who may find the original papers or other monographs too condensed or reliant on external references. The text emphasizes that while the "elementary" part (exponents equal to 2) is not necessarily easy, it is distinct from the "non-elementary" part, which requires deep tools from algebraic number theory. The work serves to validate the 2004 proof by Preda Mihăilescu, confirming that 8 and 9 are indeed the only consecutive perfect powers.

1. The Easy Cases (The "Low Hanging Fruit")

2. The Hard Part (The "Mountain Peak")

3. The Final Knockout (The "M4" Theorem)

Summary

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