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3 results

  • 3 - Quadratic Forms

  • John Stillwell, University of San Francisco
  • Book:
    Algebraic Number Theory for Beginners
    Published online:
    28 July 2022
    Print publication:
    11 August 2022, pp 59-77

  • Summary

    Solving quadratic Diophantine equations amounts to finding the values taken by quadratic forms, a problem that can be fruitfully approached by finding the equivalents of a given form under change of variables. This approach was initiated by Lagrange and developed to a high level by Gauss. However, the way Gauss did it involved an apparently difficult operation called composition of forms, clarified only later by the concept of Abelian group.

  • 2 - Diophantine Arithmetic

  • John Stillwell, University of San Francisco
  • Book:
    Algebraic Number Theory for Beginners
    Published online:
    28 July 2022
    Print publication:
    11 August 2022, pp 33-58

  • Summary

    Diophantine equations are polynomial equations for which integer (or sometimes rational) solutions are sought. The oldest examples date from ancient Greek times, and Diophantus in particular solved many such equations. His methods and the questions they raised inspired much of modern number theory, beginning with the work of Fermat and Euler. Euler, and later Gauss, introduced algebraic integers to solve Diophantine equations, implicitly or explicitly using "unique prime factorization" to do so.

  • The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields

  • Part of
  • Nuno Freitas, Samir Siksek
  • Journal:
    Compositio Mathematica / Volume 151 / Issue 8 / August 2015
    Published online by Cambridge University Press:
    06 March 2015, pp. 1395-1415
    Print publication:
    August 2015
    • Article
      • You have access
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  • Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation

    $$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$
    are the trivial ones satisfying $abc=0$. With the help of modularity, level lowering and image-of-inertia comparisons, we give an algorithmically testable criterion which, if satisfied by $K$, implies the asymptotic Fermat’s Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $K=\mathbb{Q}(\sqrt{d})$ for a subset of $d\geqslant 2$ having density ${\textstyle \frac{5}{6}}$ among the squarefree positive integers. We can improve this density to $1$ if we assume a standard ‘Eichler–Shimura’ conjecture.