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The Diophantine Equation α m + β m + γ m = 0

In: The Theory of Algebraic Number Fields

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  • David Hilbert

Abstract

Fermat advanced the conjecture that the equation $${a^m} + {b^m} + {c^m} = 0$$ is not solvable in nonzero rational integers a, b, c if m > 2. Although there were already remarkable isolated results about this Fermat equation before the time of Kummer (Abel (1), Cauchy (1, 2). Dirichlet (1, 2, 3), Lamé (1, 2, 3), Lebesgue (1, 2, 3)) nevertheless Kummer, using the theory of ideals in regular cyclotomic fields, was the first to succeed in completely proving Fermat’s conjecture for a very extensive class of exponents m. The most important result obtained by Kummer is as follows.

Suggested Citation

  • David Hilbert, 1998. "The Diophantine Equation α m + β m + γ m = 0," Springer Books, in: The Theory of Algebraic Number Fields, chapter 36, pages 327-333, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03545-0_36
    DOI: 10.1007/978-3-662-03545-0_36
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