Polynomials generalizing binomial coefficients and their application to the study of Fermat's last theorem |
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Authors: | F Thaine |
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Institution: | Departamento de Matemática, Universidade de Brasilia, Agence Postal 15, 70.000 Brasília, D.F. Brazil |
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Abstract: | We study properties of the polynomials φk(X) which appear in the formal development Πk ? 0n (a + bXk)rk = Σk ≥ 0φk(X) ar ? kbk, where rk ∈ and r = Σrk. this permits us to obtain the coefficients of all cyclotomic polynomials. Then we use these properties to expand the cyclotomic numbers Gr(ξ) = Πk = 1p ? 1 (a + bξk)kr, where p is a prime, ξ is a primitive pth root of 1, a, b ∈ and 1 ≤ r ≤ p ? 3, modulo powers of ξ ? 1 (until (ξ ? 1)2(p ? 1) ? r). This gives more information than the usual logarithmic derivative. Suppose that . Let . We prove that Gr(ξ) ≡ cp mod p(ξ ? 1)2 for some c ∈ , if and only if Σk = 1p ? 1kp ? 2 ? rmk ≡ 0 (mod p). We hope to show in this work that this result is useful in the study of the first case of Fermat's last theorem. |
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