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On Kummer's Conjecture
, 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. ..."
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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.
The prime factors of Wendt's Binomial Circulant Determinant
 Math. Comp
, 1991
"... : 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 comp ..."
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: 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...
Analysis of the classical cyclotomic approach to Fermat′s Last Theorem, Publ. Math. de Besançon, Algèbre et Théorie des Nombres, Actes de la conférence “Fonctions L et arithmétique”, Besançon 2009. Presses Universitaires de FrancheComte
, 2010
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Values of the Euler φfunction not divisible by a given odd prime, and the distribution of . . .
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A GENERALISED KUMMER’S CONJECTURE
"... 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 ElliottHalberstam Conjecture implies that this Generali ..."
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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 ElliottHalberstam Conjecture implies that this Generalised Kummer’s Conjecture is true for almost all n but is false for infinitely many n. 1.
Unit Fractions and the Class Number of a Cyclotomic Field
"... . We further examine Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, h 1 (p). Whereas Kummer had conjectured that h 1 (p) G(p) := 2p(p=4 2 ) p\Gamma1 4 we show, under certain plausible assumptions, that there ..."
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. We further examine Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, h 1 (p). Whereas Kummer had conjectured that h 1 (p) G(p) := 2p(p=4 2 ) p\Gamma1 4 we show, under certain plausible assumptions, that there exist constants a ff ; b ff such that h 1 (p) ffG(p) for a ff x= log b ff x primes p x whenever log ff is rational. On the other hand, there are A x= log A x such primes when log ff is irrational. Under a weak assumption we show that there are roughly the conjectured number of prime pairs p; mp \Sigma 1 if and only if there are AEm x= log 2 x primes p x for which h 1 (p) e \Sigma1=2m G(p). 1. Introduction Let h(p) be the class number of the cyclotomic field Q(i p ) (where i p is a primitive pth root of unity) and h 2 (p) be the class number of the real subfield Q(i p + i \Gamma1 p ). Kummer proved that the ratio h 1 (p) = h(p)=h 2 (p) is an integer which he called the first...
RELATIVE CLASS NUMBER OF IMAGINARY ABELIAN FIELDS OF PRIME CONDUCTOR BELOW 10000
"... Abstract. In this paper we compute the relative class number of all imaginary Abelian fields of prime conductor below 10000. Our approach is based on a novel multiple evaluation technique, and, assuming the ERH, it has a running time of O(p 2 log 2 (p)loglog(p)), where p is the conductor of the fiel ..."
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Abstract. In this paper we compute the relative class number of all imaginary Abelian fields of prime conductor below 10000. Our approach is based on a novel multiple evaluation technique, and, assuming the ERH, it has a running time of O(p 2 log 2 (p)loglog(p)), where p is the conductor of the field. 1.