Polynomials generalizing binomial coefficients and their application to the study of Fermat's last theorem (original) (raw)

We study properties of the polynomials Ok(X) which appear in the formal development 1 I;-o (a + bX')'" = xkiO tir(X) ar '6'. where rk E L' and r = 1 ri. This permits us to obtain the coefftcients of all cyclotomic polynomials. Then we use these properties to expand the cyclotomic numbers G,(i) = rIf : (a + bc")"'. where p is a prime, l is a primitive pth root of I. a, b E ,' and 1 <r <p-3. modulo powers of c-I (until (< ~ I)""-" r). This gives more information than the usual logarithmic derivative. Suppose that pkab(a + b). Let m =-b/a. We prove that G,(T) = cp modp(<-I)I for some CE P, if and only if \'[=I 1 kP-2 rmk E 0 (modp). We hope to show in this work that this result is useful in the study of the first case of Fermat's last theorem. Let p be a prime number, p > 5, and r be a primitive pth root of the unity. In this introduction we mention certain facts on Q(r) in connection with Fermat's Last Theorem. They led us to search for a more adequate expansion of the cyclotomic number j', (a+b{k)kr, a,bEJ, rEN. In Section I we give a generalization of Newton's Theorem on Binomial Expansion by studying the properties of the polynomials #Jx) such that /I; (a + bXk)Q = 1' qdk(X) a' kbh, ky0