共查询到20条相似文献,搜索用时 31 毫秒
1.
Let Ed(x) denote the “Euler polynomial” x2+x+(1−d)/4 if and x2−d if . Set Ω(n) to be the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariantOnod of is defined to be except when d=−1,−3 in which case Onod is defined to be 1. Finally, let hd=hk denote the class number of K. In 2002 J. Cohen and J. Sonn conjectured that hd=3⇔Onod=3 and is a prime. They verified that the conjecture is true for p<1.5×107. Moreover, they proved that the conjecture holds for p>1017 assuming the extended Riemann Hypothesis. In this paper, we show that the conjecture holds for p?2.5×1013 by the aid of computer. And using a result of Bach, we also proved that the conjecture holds for p>2.5×1013 assuming the extended Riemann Hypothesis. In conclusion, we proved the conjecture is true assuming the extended Riemann Hypothesis. 相似文献
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John Brillhart 《Journal of Number Theory》2004,106(1):79-111
Using the theory of elliptic curves, we show that the class number h(−p) of the field appears in the count of certain factors of the Legendre polynomials , where p is a prime >3 and m has the form (p−e)/k, with k=2,3 or 4 and . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y2+αxy+y=x3 and find an elementary expression for the supersingular polynomial ssp(x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(−p) also shows up in the factorization of certain Jacobi polynomials. 相似文献
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Zhi-Hong Sun 《Journal of Number Theory》2007,124(1):62-104
Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u2−dv2 and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms. 相似文献
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Let −D<−4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of exists. Let d be a fundamental discriminant prime to D. Let 2k−1 be an odd natural number prime to the class number of . Let χ be the twist of the (2k−1)th power of a canonical Hecke character of by the Kronecker's symbol . It is proved that the vanishing order of the Hecke L-function L(s,χ) at its central point s=k is determined by its root number when , where the constant implied in the symbol ? depends only on k and ?, and is effective for L-functions with root number −1. 相似文献
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Zhi-Hong Sun 《Journal of Number Theory》2008,128(5):1295-1335
Let be a prime. Let a,b∈Z with p?a(a2+b2). In the paper we mainly determine by assuming p=c2+d2 or p=Ax2+2Bxy+Cy2 with AC−B2=a2+b2. As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=bUn+k2Un−1(n?1). 相似文献
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Zhi-Hong Sun 《Journal of Number Theory》2009,129(3):499-550
Let be a prime and a,b∈Z with a2+b2≠p. Suppose p=x2+(a2+b2)y2 for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U0=0, U1=1 and Un+1=bUn+k2Un−1(n?1). We also pose many conjectures concerning , or . 相似文献
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P. Meijer 《Journal of Number Theory》2003,100(2):381-395
Let χ be a character of order 2t on the splitting field of the polynomial x2t−1 over the prime field with p≡3,5 (mod 8). In this paper we evaluate explicitly the Gauss sums related to the group of characters generated by χ. 相似文献
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Robert L. Griess Jr. 《Journal of Number Theory》2010,130(3):680-695
Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes-Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks 2d−1±2d−k−1, for odd integers d?3 and integers . Their minimum norms are moderately high: . 相似文献
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Alan Haynes 《Journal of Number Theory》2003,98(1):89-104
In this paper we examine the subset of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of the values taken on by Δ=qa′−aq′ where a/q<a′/q′ are consecutive in . After proving an asymptotic result for these frequencies, we generalize the result to the subset of elements of formed by restriction to a subinterval [α,β]⊆[0,1]. 相似文献
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Sankar Sitaraman 《Journal of Number Theory》2003,99(1):29-35
Let p>5 be a prime number and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1,(k,p)=1.Let Cp(i) be the eigenspace of the p-Sylow subgroup of ideal class group C of corresponding to ωi,ω being the Teichmuller character.In this article we extend the main theorem in Sitaraman (J. Number Theory 80 (2000) 174) and get the following: For any fixed odd positive integer n<p−4, assume:
- (a)
- At least one of Cp(3),Cp(5),…,Cp(n) is non-trivial.
- (b)
- Cp(i)=0 for p−n−1?i?p−2.
- (c)
- for 1?i?n+1.
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M.R. Bush 《Journal of Number Theory》2003,100(2):313-325
The p-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group for with d=−445,−1015,−1595,−2379. In each case we are able to narrow the identity of G down to one of a finite number of explicitly given finite groups. From this follow several results regarding the corresponding 2-class tower. 相似文献
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Zhi-Hong Sun 《Journal of Number Theory》2005,113(1):10-52
Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let εd be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that εd is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {un(a,b)} is the Lucas sequence defined by and . 相似文献
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Benoît Rittaud 《Journal of Number Theory》2007,122(2):261-282
We extend the results of uniform distribution modulo 1 given in [B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, Bull. Soc. Math. France 128 (2000) 451-471; B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, II: Cas Liouvillien unidimensionnel, Colloq. Math. 96 (1) (2003) 55-73], which deal with sequences of the form , where n(hn), and are polynomially increasing sequences, n(εn) a bounded sequence, essentially a C3-function Zd-periodic, Θ an element of Rd and t a real number. We remove the Diophantine hypothesis on Θ needed in [the first of above mentioned articles], and add a technical hypothesis on hn. We apply this result to the convergence of diagonal averages for d×d matrices. 相似文献
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