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71.
In this paper we prove that for fixed positive integers α, β, m and m ≥ 2 the equation (?) holds for n = 2 only when m = 2, α = 3 and β = 1, which is another improvement of LeVeque's result, different from Yu's. 相似文献
72.
研究丢番图方程x~y+y~z+z~x=0的可解性,并求该方程的所有整数解.本文利用初等方法及整数的整除性质研究这一问题,获得了彻底解决.即就是证明了方程x~y+y~z+z~x=0有且仅有六组整数解(x,y,z)=(-2,1,1),(1,-2,1),(1,1,-2),(1,-1,-2),(-1,-2,1),(-2,1,-1) 相似文献
73.
一个素数,两个素数的平方以及2的若干次幂和的丢番图逼近 总被引:2,自引:2,他引:0
在给定条件下证明了不等式|λ1p1 λ2p22 λ3p23 μ12x1 … μs2xs γ|<η有无限多素数p1,p2,p3和正整数x1,…,xs解. 相似文献
74.
This paper proposes three new attacks. In the first attack we consider the class of the public exponents satisfying an equation e X-N Y +(ap~r+ bq~r)Y = Z for suitably small positive integers a, b. Applying continued fractions we show thatY/Xcan be recovered among the convergents of the continued fraction expansion of e/N. Moreover, we show that the number of such exponents is at least N~(2/(r+1)-ε)where ε≥ 0 is arbitrarily small for large N. The second and third attacks works upon k RSA public keys(N_i, e_i) when there exist k relations of the form e_ix-N_iy_i +(ap_i~r + bq_i~r )y_i = z_i or of the form e_ix_i-N_iy +(ap_i~r + bq_i~r )y = z_i and the parameters x, x_i, y, y_i, z_i are suitably small in terms of the prime factors of the moduli. We apply the LLL algorithm, and show that our strategy enables us to simultaneously factor k prime power RSA moduli. 相似文献
75.
设a是大于1的正整数,f(a)是a的非负整系数多项式,f(1)=2rp+4,其中r是大于1的正整数,p=2~l-1是Mersenne素数.本文讨论了方程(a-1)x~2+f(a)=4a~n的正整数解(x,n)的有限性,并且证明了:当f(a)=91a+9时,该方程仅当a=5,7和25时分别有解(x,n)=(3,3),(11,3)和(3,4). 相似文献
76.
77.
This paper begins with a short historical survey on Catalan's equation, namely xp-yq=1, where p andq are prime numbers and x, y are non-zero rational integers. It is conjectured that the only solution is the trivial solution 32-23=1. We prove that there is no non-trivial solution with p orq smaller than 30000. The tools to reach such a result are presented. A crucial role is played by a recent estimate of linear forms in two logarithms obtained by Laurent, Mignotte and Nestrenko. The criteria used are also quite recent. We give information on the enormous amount of computation needed for the verification. 相似文献
78.
Cs. Rakaczki 《Periodica Mathematica Hungarica》2004,49(2):119-132
As a generalization of the results of [3] and [22], we characterize those pairs (m,n) and those polynomials b{Z}[x] of prime degree for which equation (1) has only finitely many integer solutions. 相似文献
79.
Rong Fang BIE Shi Qiang WANG 《数学学报(英文版)》2006,22(5):1549-1556
In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don't. 相似文献
80.
In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2+b^2=c^r,a〉b,a≡3 (mod4),b≡2 (mod4) and c-1 is not a square, thena a^x+b^y=c^z has only the positive integer solution (x, y, z) = (2, 2, r).
Let m and r be positive integers with 2|m and 2 r, define the integers Ur, Vr by (m +√-1)^r=Vr+Ur√-1. If a = |Ur|,b=|Vr|,c = m^2+1 with m ≡ 2 (mod 4),a ≡ 3 (mod 4), and if r 〈 m/√1.5log3(m^2+1)-1, then a^x + b^y = c^z has only the positive integer solution (x,y, z) = (2, 2, r). The argument here is elementary. 相似文献
Let m and r be positive integers with 2|m and 2 r, define the integers Ur, Vr by (m +√-1)^r=Vr+Ur√-1. If a = |Ur|,b=|Vr|,c = m^2+1 with m ≡ 2 (mod 4),a ≡ 3 (mod 4), and if r 〈 m/√1.5log3(m^2+1)-1, then a^x + b^y = c^z has only the positive integer solution (x,y, z) = (2, 2, r). The argument here is elementary. 相似文献