共查询到20条相似文献,搜索用时 62 毫秒
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设D1、D2、m、x、y是适合D1〉1,D2〉1,2├D1D2,gcd(D1,D2)=gcd(x,y)=1的正整数,n是适合n├h的奇素数,其中h是虚二次域Q(√-2^mD1D2)的类数。本文主要证明了:方程D1x^2+2^mD2=y^n至多有5.10^16组例外解(D1,D2,x,y,m,n)而且这些解都满足了7≤n〈8.5.10^6以及y^n〈exp(exp(exp46))。 相似文献
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qq±1(q=pn)的Aurifeuilian分解孙琦洪绍方(四川大学数学系)摘要设素数p≡ε(mod4),其中ε=1,-1,n为正整数,q=pn,q1=qq/p,η=ηq=exp(2πi/q).Φm(x)表示m阶分圆多项式.记Sε=Φq(εq),本... 相似文献
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一类表示伪素数的公式 总被引:4,自引:2,他引:2
素数最基本的性质是费马小定理,给出了自然数是素数的必要条件:若(p,a)=1(p为素数)则ap-1≡1(modp).很长一段时间以来,人门认为费马小定理的逆定理也成立,甚至认为n是素数当且仅当2n-1≡1(modn),但这是错误的.1819年萨吕斯(M.Sarrus)证明,2341≡2(mod341),但341=11×31是合数.后来,人们把满足同余式2n-1≡(modn)的合数叫伪素数.伪素数是否有无穷多?1903年,马洛(Malo)首先证明:如果A是伪素数,2A-1也是伪素数[1].文[2]给出一个伪素数的公式,笔者认为可以给出一类伪素数的公式.现给出预备知识(p为奇素… 相似文献
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“mp2型“伪素数的性质与存在 总被引:1,自引:1,他引:0
定义 若n是合数,且2n-1=1(mod n),则称n是伪素数. 文[1]证得 10932及 35112这两个数是伪素数,从而否定了陈历功等提出的“伪素数不含平方数因数”的猜想.记p是奇素数,mN,本文将讨论“mP2型”伪素数的性质与存在的实例.先引入以下的 引理[2]设使同余式:2r=1(mod m)成立的最小正整数为r,则 2a=1(mod m)的充要条件是r(注引理即文[2]第七章定理1的推过2) 定理1 设p是奇素数,如果n是含有因数P2的伪素数,则P2是伪素数. 证明 记n=mp2(m N),则由伪… 相似文献
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运用数论和图论技巧,得到了当λ(D)3时本原有向图D的广义指数exp(D,k)的界,这里λ(D)表示D中不同长的圈的类数,还证明了对任何整数n,t,不存在n阶本原有向图D,使得n2-tn+14(t+1)2+k-2<exp(D,k)<n2-(t-1)n+t+k-3. 相似文献
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设p=5(mod 6)为素数.证明了丢番图方程χ^3一У^6=3pz^2。在p=5(mod 12)为素数时均无正整数解;在P=11(mod 12)为素数时均有无穷多组正整数解,并且还获得了该方程全部正整数解的通解公式,同时还给出了该方程的部分整数解. 相似文献
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M. A. Cherepnev 《Mathematical Notes》2006,80(5-6):863-867
The main result of this paper is the fact that the fraction of primes p ≤ x satisfying the condition that p ? 1 has a prime divisor q > exp(ln x/ln ln x) and the number of prime divisors of q ? 1 essentially differ from ln ln(x/n), where n = (p ? 1)/q, tends to zero as x increases. 相似文献
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Given an integer N?≥?3, we shall prove that for all primes p?≥?(N???2)24 N , there exists x in (?/p?)* such that x, x?+?1, ..., x?+?N???1 are all squares (respectively, non-squares) modulo p. Similarly, for an integer N?≥?2, we prove that for all primes $\displaystyle p \geq \exp(2^{5.54N})$ , there exists an element x?∈?(?/p?)* such that x, x?+?1, ..., x?+?N???1 are all generators of (?/p?)*. 相似文献
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There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable replacement. In this paper, we show that the behavior of strong Krull primes most closely resembles that of associated primes over a Noetherian ring. We prove an analogue of a theorem of Epstein and Yao characterizing flat modules in terms of associated primes by replacing them with strong Krull primes. Also, we partly generalize a classical equational theorem regarding flat base change and associated primes in Noetherian rings. That is, when associated primes are replaced by strong Krull primes, we show containment in general and equality in many special cases. One application is of interest over any Noetherian ring of prime characteristic. We also give numerous examples to show that our results fail if other popular generalizations of associated primes are used in place of strong Krull primes. 相似文献
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Barry J Powell 《Journal of Number Theory》1980,12(2):210-217
While it has already been demonstrated that the set of twin primes (primes that differ by 2) is scarce in the (all twin primes) converges whereas (all primes) diverges, this paper proves in Theorems 1 and 2 the scarcity of twin primes (and, in general, of primes p which differ by any even integer as well as primes p for which yp + z is prime, y positive, z nonzero, (y, z) = 1) in a novel and natural way — by showing that the natural density of such primes compared to the set of all primes is 0, that is, , where π′(n) is the number of, say, twin primes between 1 and n for any n, and π(n) is the number of all primes between 1 and n. Theorem 3 then establishes that if a set of primes is scarce in the sense that the sum of the reciprocals of such primes converges, they are also scarce in the natural density sense outlined above. 相似文献
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<正> V.Brun最初在1920年證明了:每一充分大的偶數可表為兩個各不超過9個素數的乘積之和.簡記之為(9,9).後來,不少數學家改進與簡化了Brun方法,因此,Brun的結果也得到相應的改進, 相似文献
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It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675-686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)). 相似文献
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Hans Riesel 《BIT Numerical Mathematics》1970,10(3):333-342
By a sieve method and the use of a computer, a search for primes, forming arithmetic series, is reported. The longest series found contained 13 primes.If the local prime density in an interval is unusually large, we say that there is a cluster of primes in the interval. Clusters of large primes are searched for by looking for repetitions of patterns of primes chosen from the beginning of the prime series. The densest large cluster found is 429 983 158 710+11, 13, 17, 19, 23, 37, 41, 43, 47, 53, and 59, with 11 primes out of 49 numbers. The average prime density this high up in the number series is one number only in about 27. 相似文献
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H. Dubner T. Forbes N. Lygeros M. Mizony H. Nelson P. Zimmermann. 《Mathematics of Computation》2002,71(239):1323-1328
In 1967 the first set of 6 consecutive primes in arithmetic progression was found. In 1995 the first set of 7 consecutive primes in arithmetic progression was found. Between November, 1997 and March, 1998, we succeeded in finding sets of 8, 9 and 10 consecutive primes in arithmetic progression. This was made possible because of the increase in computer capability and availability, and the ability to obtain computational help via the Internet. Although it is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression, it is very likely that 10 primes will remain the record for a long time.
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<正> 引言 本文之目的是在證明作者在[1]內所提及的若干結果,本文所有的結果,均在廣義的Riemann猜測之下,而獲得的. 現在,先將廣義的Riemann猜測述於下: 相似文献