Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Another proof for Fermat's last theorem

      6 May, 2005
      06/05/2005

      fermat

      Fermat's last theorem says that this equation can't be satisfied for n greater than 2

      Chandrashekhar Khare, a mathematician from the University of Utah, has announced that he has proved what is known to experts as the "level-one Serre conjecture". This conjecture was posed in 1972 by the Fields medallist Jean-Pierre Serre, and belongs to the field of Arithmetic Algebraic Geometry. At this point we should of course say what the conjecture states, and explain its importance in the real world. This, however, is easier said than done, as those of us who are not experts would have to spend several months studying seriously advanced mathematics, just in order to understand the statement.

      Luckily though, the conjecture belongs in a wider context - which we can attempt to explain - and has a famous relative: Fermat's last theorem. Serre's conjecture is in a sense a parent of Fermat's last theorem: mathematicians have known for some time that if the first is true then so is the second. In fact, it is a certain part of the conjecture which implies Fermat's last theorem, and this part was proved by Khare and his collaborator J.P. Wintenberger, and independently by the mathematician Dieulefait.

      Fermat's last theorem states that if n is a whole number strictly bigger than 2, then you can never find three whole numbers x, y and z such that xn+yn=zn.

      Andrew Wiles, the man who tamed Fermat.

      Andrew Wiles, the man who tamed Fermat.

      This seems simple and has tempted many people to retire to a corner with paper and pencil and try to prove it. It is, however, devilishly hard and it wasn't until 1995 that Andrew Wiles, with the help of Richard Taylor, famously cracked the problem. Although the theorem is primarily a theorem about numbers, thus naturally belonging to the realm of number theory, to prove it Wiles had to wheel up heavy machinery from two other areas of maths: algebra and geometry.

      The fact that algebra, geometry and number theory are related can be seen using very simple maths: circles and straight lines, which are geometric objects, can be expressed by algebraic equations. If or where a circle meets a straight line can be determined by solving a set of algebraic equations, which is of course also an exercise in number theory.

      But it isn't just that algebra and geometry were used to solve Fermat's last theorem. In fact it, as well as the conjecture by Serre, are ingredients of a wider program to unify various areas of mathematics, known as Langlands philosophy. The idea behind such unifying theories is that it should be possible to directly translate every concept in a given area of maths into all the other areas of maths. Every object and concept in algebra, for example, should have a counterpart in geometry, and vice versa, and the relationship between the objects should be the same in both areas. For example, a circle and a line both have algebraic equations, and the circle and the line meeting in one, two or no points corresponds to a certain equation having one, two or no solutions.

      The Langlands philosophy was conceived by the mathematician Robert Langland in the 1960's and consists of a set of conjectures concerning the intimate relationship between number theory, geometry and algebra. What both Wiles and Khare have done is to prove bits of these conjectures, which in turn imply Fermat's last theorem.

      Khare's result came as a bit of a surprise, as no one was expected to solve this problem so soon. It is currently undergoing the meticulous scrutiny of fellow mathematicians, who will hopefully declare it fit for publication. Serre himself is said to be "happy and excited".

      Further Reading

      • You can find out about the history of Fermat's last theorem in the Plus article How maths can make you rich and famous: Part II, on Wikipedia or on MathWorld.
      • The connections between number theory and other areas of maths are well explained in the Mathematical Atlas.
      • Read about Serre's work in the Plus article En-Abeled.
      • If you're really daring you can have a look at Khare's paper on the maths archive.
      Read more about...
      Fermat's Last Theorem
      • Log in or register to post comments

      Read more about...

      Fermat's Last Theorem
      University of Cambridge logo

      Plus is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms