First case of Fermat's Last Theorem

The first case of Fermat's last theorem says that for three integers x, y and z and a prime number p, where p does not divide the product xyz, there are no solutions to the equation x^p + y^p + z^p = 0.

Using the Theorem of unique factorization of ideals in Q(ξ) it was shown that if the first case has solutions x, y, z, then x+y+z is divisible by p and (x, y), (y, z) and (z, x) are elements of H_p, where H_p denotes a set of pairs of integers with special properties.^[1]

Notes

1. Granville, A.; Monagan, M. B. (1988), "The First Case of Fermat's Last Theorem is true for all prime exponents up to 714,591,416,091,389", Transactions of the American Mathematical Society, 306 (1): 329–359, doi:10.1090/S0002-9947-1988-0927694-5.

References

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• Cikánek, P. (1994), "A Special Extension of Wieferich's Criterion" (PDF), Math. Comp., AMS, 62 (206): 923–930, doi:10.2307/2153550, JSTOR 3562296.
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• Lehmer, D. H.; Lehmer, E. (1941), "On the first case of Fermat's last theorem", Bull. Amer. Math. Soc., 47 (2): 139–142, doi:10.1090/s0002-9904-1941-07393-3
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• Jha, V. (1994), "On Krasnér's theorem for the first case of Fermat's last theorem" (PDF), Colloqium Mathematicum, 67: 25–31