ERDO S等于1987年曾证明了:对于正整数a,b,如果对所有素数p,a,b被p除所得余数分别为a(m od p),b(m od p),都有a(m od p)≤b(m od p),则a=b.该文则研究对哪些正整数a,b,满足对所有素数p,恒有a(m od p)≤b(m od p)+1,对1≤a≤5,确定了所有...ERDO S等于1987年曾证明了:对于正整数a,b,如果对所有素数p,a,b被p除所得余数分别为a(m od p),b(m od p),都有a(m od p)≤b(m od p),则a=b.该文则研究对哪些正整数a,b,满足对所有素数p,恒有a(m od p)≤b(m od p)+1,对1≤a≤5,确定了所有的b.即当a=1时,b可取一切正整数;a=2时,b=2k,k=0,1,2,…;a=3时,b=2,3,4,9;a=4时,b=3,4;a=5时,b=4,5.展开更多
Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizatio...Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.展开更多
Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expre...Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expressible by the sum of a prime number and an element of u has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form p ＋ [（2 ＋ √3）n], where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be ＂too small＂ for most prime numbers p.展开更多
目的 随着存在大量低性能电子设备的物联网系统迅速发展和普及,人们对低精度计算环境下安全高效的图像加密技术有着越来越迫切的需求。现有以混沌系统为代表的图像加密方法不仅加密速度普遍较低,而且在低精度计算环境下存在严重的安全缺...目的 随着存在大量低性能电子设备的物联网系统迅速发展和普及,人们对低精度计算环境下安全高效的图像加密技术有着越来越迫切的需求。现有以混沌系统为代表的图像加密方法不仅加密速度普遍较低,而且在低精度计算环境下存在严重的安全缺陷,难以满足实际需求。针对上述问题,本文提出了一种基于素数模乘线性同余产生器的批图像加密方法,用以提升低精度环境下图像加密的效率和安全性。方法 该方法的核心是构建一个能在低精度环境下有效运行的素数模乘线性同余产生器;将图像集均分为3组,并借助异或运算生成3幅组合图像;接着引入图像集的哈希值更新上述第3组图像;将更新后的组合图像作为上述产生器的输入,进而生成一个加密序列矩阵;基于加密序列矩阵对明文图像进行置乱和扩散,并使用异或运算生成密文图像;使用具有较高安全性的改进版2D-SCL(a new 2D hypher chaotic map based on the sine map, the chebysher map and a linear function)加密方法对加密序列矩阵进行加密。结果 仿真结果表明,本文提出的批图像加密方法在计算精度为2-8的情况下不仅能抵御各类攻击,而且加密速度相较于对比加密方法有所提升。而对比加密方法在上述计算精度环境下存在不能抵御相应攻击的情况。结论 本文提出的基于素数模乘线性同余产生器的批图像加密方法,不仅有效地解决了低计算精度环境下图像加密安全性低的问题,而且还大幅提升了图像的加密速度,为后续高效安全图像加密方法的研究提供了一个新的思路。展开更多
This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1（mod 4）or a〉1then [3/4pa]∑...This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1（mod 4）or a〉1then [3/4pa]∑k=0≡（2/pa）（mod p^2）where（—）denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that p-1∑k=1 Lk/k^2≡0（mod p） provided p〉5.where the Lucas numbers Lo,L1,L2,...are defined by L_0=2,L1=1 and Ln＋1=Ln＋Ln-l（n=1,2,3,...）.The third theorem states that if p=5 then Fp^a-（p^a/5）mod p^3 can be determined in the following way： p^a-1∑k=0（-1）^k（2k k）≡（p^a/5）（1-2F p^a-（pa/5））（mod p^3）which appeared as a conjecture in a paper of Sun and Tauraso in 2010.展开更多
文摘ERDO S等于1987年曾证明了:对于正整数a,b,如果对所有素数p,a,b被p除所得余数分别为a(m od p),b(m od p),都有a(m od p)≤b(m od p),则a=b.该文则研究对哪些正整数a,b,满足对所有素数p,恒有a(m od p)≤b(m od p)+1,对1≤a≤5,确定了所有的b.即当a=1时,b可取一切正整数;a=2时,b=2k,k=0,1,2,…;a=3时,b=2,3,4,9;a=4时,b=3,4;a=5时,b=4,5.
文摘研究了一类不定方程求正整数解的问题.借助一个引理,推导并证明了不定方程x2-py2=z2(p为奇素数)正整数解的一般公式.不定方程x2-py2=z2(p为奇素数)满足(x,y)=1的一切正整数解可表示为x=12(a2+pb2),y=ab,z=12a2-pb2,这里a>0,b>0,a,b都是奇数,p a;或x=a2+pb2,y=2ab,z=a2-pb2,这里a>0,b>0,a,b一奇一偶,p a.
文摘Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.
文摘Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expressible by the sum of a prime number and an element of u has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form p ＋ [（2 ＋ √3）n], where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be ＂too small＂ for most prime numbers p.
文摘目的 随着存在大量低性能电子设备的物联网系统迅速发展和普及,人们对低精度计算环境下安全高效的图像加密技术有着越来越迫切的需求。现有以混沌系统为代表的图像加密方法不仅加密速度普遍较低,而且在低精度计算环境下存在严重的安全缺陷,难以满足实际需求。针对上述问题,本文提出了一种基于素数模乘线性同余产生器的批图像加密方法,用以提升低精度环境下图像加密的效率和安全性。方法 该方法的核心是构建一个能在低精度环境下有效运行的素数模乘线性同余产生器;将图像集均分为3组,并借助异或运算生成3幅组合图像;接着引入图像集的哈希值更新上述第3组图像;将更新后的组合图像作为上述产生器的输入,进而生成一个加密序列矩阵;基于加密序列矩阵对明文图像进行置乱和扩散,并使用异或运算生成密文图像;使用具有较高安全性的改进版2D-SCL(a new 2D hypher chaotic map based on the sine map, the chebysher map and a linear function)加密方法对加密序列矩阵进行加密。结果 仿真结果表明,本文提出的批图像加密方法在计算精度为2-8的情况下不仅能抵御各类攻击,而且加密速度相较于对比加密方法有所提升。而对比加密方法在上述计算精度环境下存在不能抵御相应攻击的情况。结论 本文提出的基于素数模乘线性同余产生器的批图像加密方法,不仅有效地解决了低计算精度环境下图像加密安全性低的问题,而且还大幅提升了图像的加密速度,为后续高效安全图像加密方法的研究提供了一个新的思路。
基金supported by National Natural Science Foundation of China(Grant Nos.10901078 and 11171140)
文摘This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1（mod 4）or a〉1then [3/4pa]∑k=0≡（2/pa）（mod p^2）where（—）denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that p-1∑k=1 Lk/k^2≡0（mod p） provided p〉5.where the Lucas numbers Lo,L1,L2,...are defined by L_0=2,L1=1 and Ln＋1=Ln＋Ln-l（n=1,2,3,...）.The third theorem states that if p=5 then Fp^a-（p^a/5）mod p^3 can be determined in the following way： p^a-1∑k=0（-1）^k（2k k）≡（p^a/5）（1-2F p^a-（pa/5））（mod p^3）which appeared as a conjecture in a paper of Sun and Tauraso in 2010.
