Abstract
In a paper from 1934, Vandiver sketched the proof of the claim that the First Case of Fermat’s Last Theorem follows from the conjecture presently bearing his name. In 1993, Sitaraman showed that the existing gap in Vandiver’s proof could easily be filled by adding a condition on the class group of the pth cyclotomic field. In this paper, we give a proof of a slightly more general result than the one of Vandiver–Sitaraman, with consequences for a larger family of Diophantine equations.
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Notes
- 1.
Note that −1 is a pth power residue, so we may disregard signs in the evaluation of the residue symbols.
References
Banaszak, G., Gajda, W.: On the arithmetic of cyclotomic fields and the K-theory of Q. In: Algebraic K-Theory. Contemp. Math., vol. 199. Amer. Math. Soc., Providence (1996)
Gras, G., Quême, R.: Some works of Furtwängler and Vandiver revisited and Fermat’s last theorem (2011). arXiv:1103.4692
Hasse, H.: Algebraische Zahlkörper, 2nd edn. Physica Verlag, Würzburg (1965)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, vol. 84. Springer, Berlin (1990)
Jha, V.: The stickelberger ideal in the spirit of Kummer with applications to the first case of Fermat’s last theorem. Queens’s Papers in Pure and Applied Mathematics 93 (1993)
Kurihara, M.: Some remarks on conjectures about cyclotomic fiels and k-groups of ℤ. Compos. Math. 81, 223–236 (1992)
Mihăilescu, P.: Class number conditions for the Diagonal case of the equation of Nagell and Ljunggren. In: Diophantine Approximation, Festschrift for W. Schmidt’s 70th Birthday, pp. 245–273. Springer, Berlin (2008)
Ribenboim, P.: 13 Lectures on Fermat’s Last Theorem. Springer, Berlin (1979)
Sitaraman, S.: Vandiver revisited. J. Number Theory 57(1), 122–129 (1996)
Soulé, C.: Perfect forms and Vandiver’s conjecture. J. Reine Angew. Math. 517, 209–221 (1999)
Vandiver, H.S.: Some theorems concerning properly irregular cyclotomic fields. Proc. Natl. Acad. Sci. USA 15, 202–207 (1929)
Vandiver, H.S.: On power characters of singular integers in a properly irregular cyclotomic field. Trans. Am. Math. Soc. 32, 391–408 (1930)
Vandiver, H.S.: Fermat’s last theorem and the second factor in the cyclotomic class number. Bull. Am. Math. Soc. 40, 118–126 (1934)
Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83. Springer, Berlin (1996)
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Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.
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Mihăilescu, P. (2012). On Vandiver’s Best Result on FLT1. In: Pardalos, P., Georgiev, P., Srivastava, H. (eds) Nonlinear Analysis. Springer Optimization and Its Applications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3498-6_25
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DOI: https://doi.org/10.1007/978-1-4614-3498-6_25
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