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The Pell Equation

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Quadratic Number Fields

Part of the book series: Springer Undergraduate Mathematics Series ((SUMS))

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Abstract

We give the classical proof of the solvability of the Pell equation, present a method for computing the fundamental unit, and show how to apply these results to the determination of squares in Lucas sequences.

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Notes

  1. 1.

    By this we mean solutions with y≠0.

  2. 2.

    See Exercise 7.1.

  3. 3.

    Strictly speaking, the investigation of Platon’s side and diagonal numbers by Theon may be seen as the only serious investigation of a Pell equation in ancient Greece. Equations of Pell type also figure prominently in the Cattle Problem of Archimedes; it is not known, however, whether there were any attempts at solving this problem before it was discovered by Lessing in 1773.

  4. 4.

    We have derived the rational parametrization of Pell conics in Theorem 3.1.

  5. 5.

    An excellent account of Indian mathematics was given by Kim Plofker [104]. For an investigation of the Indian method of solving the equation Nx 2 + 1 = y 2, see [114].

  6. 6.

    It seems that this principle was given a name rather late (in the twentieth century?); a pigeonhole is a drawer, so the last thing you would like to put there are pigeons.

  7. 7.

    Good sources for the state of the art are [20, 91], and, in particular, [66].

  8. 8.

    This theorem is due to Halter-Koch [48] and the proof presented here to Mollin [95].

  9. 9.

    The class number formula roughly implies that fields with large fundamental units tend to have small class numbers; constructing families of fields with large fundamental units is therefore important with respect to Gauss’s conjecture that there are infinitely many real quadratic number fields with class number 1.

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Authors and Affiliations

  1. Jagstzell, Germany

    Franz Lemmermeyer

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  1. Franz Lemmermeyer
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Lemmermeyer, F. (2021). The Pell Equation. In: Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-78652-6_7

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