: Wm , Wendt's binomial circulant determinant, is the determinant of an m by m circulant matrix of integers, with (i; j)th entry i m ji\Gammajj j whenever 2 divides m but 3 does not. We explain how we found the prime factors of Wm for each even m 200 by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and q = mp + 1 are odd primes, 3 does not divide m and m 200, then the first case of Fermat's Last Theorem is true for exponent p. 1. Introduction. For a given positive even integer m, define Wm to be the determinant of the m by m circulant matrix with top row (a 0 ; a 1 ; : : : ; am\Gamma1 ) where gm (X) := m\Gamma1 X i=0 a i X i := 8 ! : (X + 1) m \Gamma X m if 6 does not divide m; (X+1) m \GammaX m (X 2 +X+1) if 6 divides m. When 6 does not divide m, the (i; j)th entry is i m ji\Gammajj j and this matrix is given the name in the title. There are a variety of app...

2001 Kummer conjectured the asymptotic behavior of the first factor of the class number of a cyclotomic field. If we only ask for upper and lower bounds of the order of growth predicted by Kummer, then this modified Kummer conjecture is true for almost all primes. 2001 Academic Press 1.

Abstract. Kummer’s Conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer’s Conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott-Halberstam Conjecture implies that this Generalised Kummer’s Conjecture is true for almost all n but is false for infinitely many n. 1.