The Pell Equations x^2-8y^2=1 and y^2-Dz^2=1

In this paper,we have proved that if one of the following conditions is satisfed,then the equations in title has no positive integer solution:①D=∏si=1P i or D=2∏si=1P i and \{ P i≡3 (mod 4)\} (1≤i≤s) or P i≡5 (mod 8) (i≤i≤s); ② D=∏si=1P i-1 (mod 12), 1≤s≤7 and \{D≠3·5·7·11·17·577,7·19·29·41·59·577;\} ③ D=2∏si=1P i,1≤s≤6 and \{D ≠2·17,2·3·5·7·11·17,2·17·113·239·337·577·665857;\} ④ D=∏si=1P i≡-1 (mod 12), 1≤s≤3 and D≠ 5·7,29·41·239.

Diophantine Approximation with Four Squares of Primes and Powers of Two

Under certain condition, the inequality |λ_1p_1~2+λ_2p_2~2+λ_3p_3~2+λ_4p_4~2+μ_12^(x1)+…+μ_s2^(xs)+γ|＜ηhas infinitely many solutions in primes p_1,p_2,p_3,p_4 and positive integers x_1,…,x_s.

On the multiplicity of binary recurrences

Let A∈N,B∈Z with gcd(A,B)=1,B{-1,0,1}. For the binary recurrence (Lucas sequence) of the form u 0=0, u 1=1, u n+2 =Au n+1 +Bu n, let N 1(A,B,k) be the number of the terms n of |u n|=k, where k∈N. In this paper, using a new result of Bilu, Hanrot and Voutier on primitive divisors, we proved that N 1(A,B,k)≤1 except N 1(1,-2,1)=5[n=1,2,3,5,13], N 1(1,-3,1)=3, N 1(1,-5,1)=3,N 1(1,B,1)=2(B{-2,-3,-5}), N 1(12,-55,1)=2, N 1(12,-377,1)=2, N 1(A,B,1)=2(A 2+B=±1, A>1), N 1(1,-2,3)=2, N 1(A,B,A)=2(A 2+2B=±1,A>1. For Lehmer sequence, we got a similar result. In addition, we also obtained some applications of the above results to some Diophantime equations.

Heron Triangle and Diophantine Equation

In this paper, we study the quantic Diophantine equation （1） with elementary geometry method, therefore all positive integer solutions of the equation （1） are obtained, and existence of Heron triangle whose median lengths are all positive integer are discussed here.

Simple groups with orders 2~a3~b5~cp^d, 2~a3~b7~cp^d and 2~a3~b5~c7~d

This work deals with the power exponent 1rand 2r respectively of the maximal and second-maximal prime factors of the order of simple K4-group, and the classification for simple 4{5,7}K--group G (i.e. G can not be divided by 5 nor by 7 or ()Gp= 4 ), simple 5 -4K-group G (i.e. G can not divided by 5 and ()Gp=4) and simple 7-4K-group G (i.e. G can not divided by 7 and ()Gp= 4). It is derived that 1r =1, 2 and 4, and 2r is not greater than 4. All the simple 4K-groups with order 235,237abcdabcdpp and 2357abcd are obtained.

On the Diophantine Equation y^2= px（x^2＋ 2）

For any fixed odd prime p, let N（p） denote the number of positive integer solutions （x, y） of the equation y^2 = px（x^2 ＋ 2）. In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7（mod 8）, then N（p） = 0; ifp ≡ 1（mod 8）, then N（p） 〈 1; if p〉 3 andp ≡ 3（rood 8）, then N（p） ≤ 2.

On the Exponential Diophantine Equation x^2 + (3a^2 -1)~m = (4a^2 -1)~n

We apply a new, deep theorem of Bilu, Hanrot ＆ Voutier and some fine results on the representation of the solutions of quadratic Diophantine equations to solve completely the exponential Diophantine equation x^2＋（3a^2-1）^m = （4a^2-1）^n when 3a^2-1 is a prime or a prime power.

对代数发展作出奠基性贡献的数学家

希腊数学家丢番图第一次把未知数引入代数,并使用了一整套符号表示未知数,使代数中数量关系的表述变得更为紧凑、有效,为代数的发展注入了活力,推动了代数学的发展及数学思想的重大变革,所著《算术》(3世纪左右)是人类历史上最早的一部代数学巨著,与欧凡里得的《几何原本》齐名,使代数脱离了几何的特征,从而使代数成为一门独立的科学.

On the solutions of a system of two Diophantine equations

We obtain all positive integer solutions（m1,m2,a,b） with a ＆gt; b,gcd（a,b） = 1 to the system of Diophantine equations km21- lat1bt2a2r= C1,km22- lat1bt2b2r= C2,with C1,C2 ∈ {-1,1,-2,2,-4,4},and k,l,t1,t2,r ∈ Z such that k ＆gt; 0,l ＆gt; 0,r ＆gt; 0,t1 ＆gt; 0,t2 0,gcd（k,l） = 1,and k is square-free.

On the diophantine equation x^2-Dy^2=n

We propose a method to determine the solvability of the diophantine equation x^2 - Dy^2 = n for the following two cases：the following two cases：（1） D = pq, where p, q ≡ 1 mod 4 are distinct primes with （q） = 1 and p q （p/q（q/p）4 = -1.（2） D = 2p1P2 ... Pro, where Pi ≡ 1 mod 8, 1 ≤ i ≤m are distinct primes and D = r2 ＋s2 with r, s ≡±3 mod 8.

SOME RESULTS ON GENERALIZED DIFFERENCE SETS

Sequences with ideal correlation functions have important applications in communications such as CDMA, FDMA, etc. It has been shown that difference sets can be used to construct such sequences. The author extends Pott and Bradley＇s method to a much broader case by proposing the concept of generalized difference sets. Some necessary conditions for the existence of generalized difference sets are established by means of some Diophantine equations. The author also provides an algorithm to determine the existence of generalized difference sets in the cyclic group Zv. Some examples are presented to illustrate that our method works.

On the size of the intersection of two Lucas sequences of distinct type Ⅱ

Let a and b be positive integers, with a not perfect square and b > 1. Recently, He, Togband Walsh proved that the Diophantine equation x2-a((bk-1)/(b-1))2=1 has at most three solutions in positive integers. Moreover, they showed that if max{a,b} > 4.76·1051, then there are at most two positive integer solutions (x,k). In this paper, we sharpen their result by proving that this equation always has at most two solutions.