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 xp + yp + zp = 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 Hp, where Hp 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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