On the Diophantine Equation x~4-Dy~2=1

The research of the Diophantine equation X~4-Dy~2=1 (1) was started by Ljunggren in 1942, where D>0 and is not a square integer. Since then thanks to the work of Ljunggren, Cohn, Bumby, and Ko Chao(柯召). Sun Chi(孙琦), many advances of the research have been made. One can refer to [1] and its references.     

On the Diophantine Equation x~4- Dy~2= 1

ＯｎｔｈｅＤｉｏｐｈａｎｔｉｎｅＥｑｕａｔｉｏｎｘ~４－Ｄｙ~２＝１ＳｕｎＱｉ（孙琦）ＴｈａｎＰｉｎｇｚｈｉ（袁平之）（ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ，Ｓｉｃｈｕａｎ，６１００６４）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＫｅＺｈａｏＲｅｃｅｉ...     

On the Diophantine Equation sum from k=1 to m((k~n)=(m+1)~n)

P. Erds has conjectured [1] that the Diophantine equation 1~n+2~n+…+m~n=(m+1)~n (1) has no positive integer solutions except that n=1, m=2. It is true when m≤10~(10) [3]. A generalized form of (1) has been investigated in [1] [2], and various     

On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

In this study, we determine when the Diophantine equation x ²?kxy+y ²?2 ⁿ = 0 has an infinite number of positive integer solutions x and y for 0 ? n ? 10. Moreover, we give all positive integer solutions of the same equation for 0 ? n ? 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x ² ? kxy + y ² ? 2 ⁿ = 0.     

On the Diophantine Equation x2 - kxy ＋ y2 ＋ 1x = 0

Let l be a given nonzero integer.The authors give an explicit characterization of the positive integer k that makes the Diophantine equation x2-kxy+y2+lx=0 have infinitely many positive integer solutions(x,y).     

On the Equation n =p k~2 in Short Intervals

ＯｎｔｈｅＥｑｕａｔｉｏｎｎ＝ｐ＋ｋ~２ｉｎＳｈｏｒｔＩｎｔｅｒｖａｌｓ￥ＷａｎｇＴｉａｎｚｅ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ４７５００１）Ａｂｓｔｒａｃｔ：ＤｅｎｏｔｅｂｙＥ（Ｘ，Ｙ）?..     

关于Diophantine方程(n+1)+(n+2)y=nz

设n是正整数.本文证明了:方程(n+1)+(n+2)y=nz仅当n=3时有正整数解(y,z)=(1,2).     

On the Matrix Equation A~2=J

An unsolved problem is to enumerate all solutions for the matrix equation A~2=Jwhere A is an n×n (0.1)-matrix and J is the n×n matr trix with every entry being 1. Here we present a family of n×n generalized circulant (0.1)-matrices each of whose square is J. Moreover. the existence of this family is unique up to permutational similarity.     

On Globally Periodic Solutions of the Difference Equation x n + 1 = f ( x n )/ x n − 1

We study the second-order difference equation x n +1 = f ( x n ) x n m 1 where f ] C 1 ([0, X ),[0, X )) and x n ] (0, X ) for all n ] Z . For the cases p h 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p . We answer some questions and conjectures of Kulenovi ' and Ladas.     

关于函数方程f1n1+af1m1f2m2+f2n2=1

对于函数方程f1n1+af1m1f2m2+f2n2=1,其中a∈C/{0},n1,n2,m1,m2∈N,给出存在非常数亚纯函数解和整函数解的必要条件.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

More on the Difference Equation y n + 1 = ( p + y n −1 )/( qy n + y n −1 )

From previous studies of the equation in the title with positive parameters p and q and positive initial conditions we know that if q h 4 p + 1 then the equilibrium is a global attractor. We also know that if q > 4 p + 1 then every solution eventually enters and remains in the interval [ p / q , 1]. In this strip there exists a "unique" prime period two solution that is locally asymptotically stable. In this paper, we provide more insight as to the behavior of solutions of the equation in the title in the strip [ p / q , 1], where a one-dimensional stable manifold lives.     

On the Prime Solutions of the Diophantine Equation (X~m-1)/(X-1)=Y~n

ＯｎｔｈｅＰｒｉｍｅＳｏｌｕｔｉｏｎｓｏｆｔｈｅＤｉｏｐｈａｎｔｉｎｅＥｑｕａｔｉｏｎ（Ｘ~ｍ－1）/（Ｘ－１）＝Ｙ~ｎＬｅＭａｏｈｕａ（乐茂华）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＺｈａｎｊｉａｎｇＴｅａｃｈｅｒｓＣｏｌｌｅｇｅ，Ｚｈａｎｊｉａｎ，Ｇｕａｎｇｄ...     

On the positive integral solutions of the Diophantine equation x 3+by+1−xyz=0

In this paper, we give a formula for the number of positive integral solutions (x,y,z) of the equation x ³+by+1?xyz=0 and also we prove a stronger form of a conjecture of S.P. Mohanty and A.M.S. Ramasamy concerning the number of positive integral solutions (x,y,z) of the equation.     

On the Generalized Hurwitz Equation and the Baragar–Umeda Equation

We consider the generalized Hurwitz equation \({a_1x_1^2 + \cdots + a_nx_n^2 = dx_1 \cdots x_n - k}\) and the Baragar–Umeda equation \({ax^2 + by^2 + cz^2 = dxyz + e}\) for solvability in integers.     

关于指数Diophantine方程x~2=D~(2m)-D~mp~n+p~(2n)

设D 1是正整数,p是适合p?D的素数.本文研究了指数Diophantine方程x~2=D~(2m)-D~mp~n+p~(2n)的满足m 1的正整数解.根据Diophantine方程的性质,结合已有的结论,运用初等方法确定了方程满足m 1的所有正整数解(D,p,x,m,n).这个结果修正并完整解决了文献[4]的猜想.