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1.
对于任意正整数n,令σ(n)表示为n的所有正因数的和函数.对于正整数n,若存在正整数m满足关系式σ(n)=σ(m)=n+m,则称正整数数对(n,m)为一对亲和数;若不存在正整数m满足关系式σ(n)=σ(m)=n+m,则称n为孤立数.亲和数与孤立数是数论中的两类重要的整数.利用初等方法结合计算机python语言,证明了整数E(33,t)=1/2(33^(2^(t))+1)是孤立数.  相似文献   

2.
关于数论函数σ(n)的一个注记   总被引:2,自引:0,他引:2  
对于两个不相同的正整数m和n,如果满足σ(m)=σ(n)=m n,则称之为一对亲和数,这里σ(n)=∑d|nd.本文给出了f(x,y)=x2x y2x(x>y≥1,(x,y)=1)不与任何正整数构成亲和数对的结论,这里x,y具有不同的奇偶性,即,关于z的方程σ(f,(x,y))=σ(z)=f(x,y) z不存在正整数解.  相似文献   

3.
运用初等方法讨论有关奇完全数的两个猜想.证明了:(i)如果n=p~αq_1~(2β_1)q_2~(2β_2)…q_s~(2β_s)是奇完全数,其中P,q_1,q_2,…,q_s是不同的奇素数,α,β_1,β_2,…,β_s是正整数,p≡α≡1(mood4),而且q_i≡-1(mod m)(i=1,2,…,s),m是大于2的正整数,则.1/2σ(p~α)必为合数;(ii)如果n=a~2~x+b~2~x,其中a,b,x是适合ab,gcd(a,6)=1,2|ab的正整数,则当x≥log_2log_2log_2 a时,n不是奇完全数.  相似文献   

4.
设ρ是可乘算术函数,定义为对每个素数方幂p~α,ρ(p~α)=p~α-p~(α-1)+p~(α-2)-…+(-1)~α.对正整数n,若2ρ(n)=n+d,其中d是n的真因子,则称n为盈因子是d的盈不完全数.本文得到了具有三个不同素因子的所有奇盈不完全数和部分偶盈不完全数.  相似文献   

5.
奇完全数的几个命题   总被引:1,自引:0,他引:1  
本文证明形如3m-1的正整数不是完全数,由此推出当所有的qi≡-1(mod 3)时,奇数n=p~αП_(i=1)~sq_i~(2β_i)若是完全数,那么1/2σ(p~α)必是合数.指出k倍完全数的素因子必须满足一个不等式.运用此不等式证明当a≥n-2时形如a~(2~n)+b~(2~n)(a>b>0,a,b,n∈N~+)的奇数不是完全数.还指出当a与b都与3互素时,对于任意的正整数m,n,奇数a~(2~n)+b~(2~m)不是完全数.  相似文献   

6.
设n为自然数,σ(n)表示n的所有正因子和函数.令d是n的真因子,若n满足σ(n)=2n-d,则称n为亏因子为d的亏完全数.本文给出了具有四个素因子的奇亏完全数的一些性质的刻画.  相似文献   

7.
给定正整数N,如果d|N且(d,N/d)=1,则称d为N的unitary因子.设k≥2为整数,r*(N)表示N的所有unitary因子的和.若σ~*(N)=kN,则称N为k重unitary完全数.本文给出了k重unitary完全数的一些性质.  相似文献   

8.
对于两个不相同的正整数$m$和$n$, 如果满足$\sigma(m)=\sigma(n)=m+n$, 则称之为一对亲和数, 这里$\sigma(n)=\sum_{d|n}d$.本文给出了$f(x,y)=x^{2^{x}}+y^{2^{x}}(x>y\geq{1},(x,y)=1)$不与任何正整数构成亲和数对的结论, 这里$x$,$y$具有不同的奇偶性, 即, 关于$z$的方程$\sigma(f(x,y))=\sigma(z)=f(x,y)+z$不存在正整数解.  相似文献   

9.
实二次城 Q(■)类数的可除性   总被引:4,自引:4,他引:0  
乐茂华 《数学学报》1990,33(4):565-574
设 d 是无平方因子正整数,h(d)是实二次域 Q(d~(1/2))的类数.本文证明了:如果 da~2=1+4k~(2n),a、k、n 是正整数,k>1,n>1,n 的奇素因子 p和 k 的素因子 q 都适合 gcd(p,(q-1)q)=1,而且 2k~n+ad~(1/2)是 Pell 方程u′~2-dv′~2=-1 的基本解,则除了(a,d,k,n)=(5,41,2,4) 以及 n=2,k=P_mP_(m+1) 或者 2Q_mQ_(m+1) 以外,h(d)=0(modn),这里 m 是正整数,P_m=1/2((1+2~(1/2))~m+(1-2~(1/2))~m),Q_m=1/22~(1/2)((1+2~(1/2))~m-(1-2~(1/2))~m).由此可推得:对于任何正整数 n,存在无限多个实二次域,可使 n 整除其类数.  相似文献   

10.
二阶三参数混合型偏差分方程解的振动性   总被引:1,自引:0,他引:1  
应用包络理论主要研究了偏差分方程pU_(m+2,n)+qU_(m,n+2)-U_(m,n)+rU_(m+σ,n-τ)=0,解的振动性,其中参数p,q,r是实数,σ,τ为正整数,m,n为非负整数.  相似文献   

11.
On the basis of the Arabic text, we investigate how Th bit ibn Qurra (ninth century) could have found his rule for amicable numbers.  相似文献   

12.
In a first article of this title, new procedures were described to compute many amicable numbers by ``breeding' them in several generations. An extensive computer search was later performed (in 1988), and demonstrated the remarkable effectiveness of this breeding method: the number of known amicable pairs was easily quadrupled by this search. As we learnt recently (1999) from the internet, Pederson and te Riele have again multiplied that number roughly by ten. While they give no information on their method of search, we publish here our method and summarize the computations. Our results provide some evidence for the conjecture that the number of amicable pairs is infinite.

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13.
素数的立方都是孤立数   总被引:2,自引:0,他引:2  
本文运用数论函数的基本性质讨论了相亲数的存在性,证明了素数的立方都是孤立数.  相似文献   

14.
康托尔实数的局限性   总被引:1,自引:0,他引:1  
罗里波 《数学研究》2008,41(1):72-78
康托尔为我们建立了集合论,并且证明了实数的不可数性,但是其中留下了很多疑点. 1.—个实数能在每—个集合论模型中出现的充分必要条件是它是可以被集合论来定义的.那些在集合论模型中不出现的实数,我们可以把他们叫做看不见的实数. 2.在实数的十进位无穷小数表示法中有些是我们能确切地知道它的第几位是什么,但是对另外的一些实数我们对它们就只能有模糊的认识,也就是说它的第几位是什么我们不可能全部知道.我们可以把他们叫做写不出的实数. 3.由于Cantor关于实数是不可数的证明不是构造性的证明,而是用所谓的归谬证法.它们中有很多是看不见写不出的实数.因此说它们是虚拟的实数. 4.虚拟实数就像银行中的虚拟货币,你可用它来买东西,它可从—个户头转拨到另—个户头,但是钱的实体是不存在的。这个现象也让我们对某些数学工具的合法性挺出质疑.我们用对角线法来证明实数的基数比自然数的基数大。但是我们并没有真正有效的地构造出那么多的实数.因此我们没有办法来确切地定义它们.也可以说它们中的绝大多数是不可以定义的.在一般的情况下虚拟实数是不可以个别地使用的.  相似文献   

15.
In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.  相似文献   

16.
Fifty new amicable four-cycles are discovered by the constructive method invented in 1969 by the second author.

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17.
We consider notions of boundedness of subsets of the natural numbers ? that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and we formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of ? to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most of the results do not require countable choice. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

18.
In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers.  相似文献   

19.
Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.  相似文献   

20.
Although it is known that the maximum number of variables in two amicable orthogonal designs of order 2np, where p is an odd integer, never exceeds 2n+2, not much is known about the existence of amicable orthogonal designs lacking zero entries that have 2n+2 variables in total. In this paper we develop two methods to construct amicable orthogonal designs of order 2np where p odd, with no zero entries and with the total number of variables equal or nearly equal to 2n+2. In doing so, we make a surprising connection between the two concepts of amicable sets of matrices and an amicable pair of matrices. With the recent discovery of a link between the theory of amicable orthogonal designs and space‐time codes, this paper may have applications in space‐time codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 240‐252, 2009  相似文献   

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