共查询到20条相似文献,搜索用时 0 毫秒
1.
WEI DaSheng 《中国科学 数学(英文版)》2013,56(2):227-238
We propose a method to determine the solvability of the diophantine equation x2-Dy2=n for the following two cases:(1) D = pq,where p,q ≡ 1 mod 4 are distinct primes with(q/p)=1 and(p/q)4(q/p)4=-1.(2) D=2p1p2 ··· pm,where pi ≡ 1 mod 8,1≤i≤m are distinct primes and D=r2+s2 with r,s ≡±3 mod 8. 相似文献
2.
Guo Xiaoyan 《Periodica Mathematica Hungarica》2013,66(1):87-93
Let a, b be fixed positive integers such that a ≠ b, min(a, b) > 1, ν(a?1) and ν(b ? 1) have opposite parity, where ν(a ? 1) and ν(b ? 1) denote the highest powers of 2 dividing a ? 1 and b ? 1 respectively. In this paper, all positive integer solutions (x, n) of the equation (a n ? 1)(b n ? 1) = x 2 are determined. 相似文献
3.
Let ?, ? be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, there have been many papers concerned with solutions (x, y, n, a, b) of the equation x 2 + 2 a p b = y n , x, y, n ε ?, gcd(x, y) = 1, n ? 3, a, b ε ?, a ? 0, b ? 0. And all solutions of it have been determined for the cases p = 3, p = 5, p = 11 and p = 13. In this paper, we mainly concentrate on the case p = 3, and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions (x, y, n, a, b) of the equation x 2+2 a · 17 b = y n , x, y, n ε ?, gcd(x, y) = 1, n ? 3, a, b ∈ ?, a ? 0, b ? 0, are determined. 相似文献
4.
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. 相似文献
5.
İsmaıl Nacı Cangül Musa Demırcı İlker İnam Florian Luca Gökhan Soydan 《Mathematica Slovaca》2013,63(3):647-659
In this note, we find all the solutions of the Diophantine equation x 2 + 2 a · 3 b · 11 c = y n , in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime. 相似文献
6.
Peter M. Knopf 《Journal of Difference Equations and Applications》2013,19(7):607-619
Consider the third-order difference equation x n+1 = (α+βx n +δx n ? 2)/(x n ? 1) with α ∈ [0,∞) and β,δ ∈ (0,∞). It is shown that this difference equation has unbounded solutions if and only if δ>β. 相似文献
7.
Let 2 ≤ p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [27], [2], [3], [5], [23].
Received: 25 April 2008 相似文献
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Let a≥1 be an integer.In this paper,we will prove the equation in the title has at most three positive integer solutions. 相似文献
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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. 相似文献
13.
S. Subburam 《The Ramanujan Journal》2013,32(2):203-219
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. 相似文献
14.
Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n 2 ? 1 and 12n 2 + 1 are odd primes, then the general elliptic curve y 2 = x 3+(36n 2?9)x?2(36n 2?5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y 2 = x 3 + 27x ? 62 really is an unusual elliptic curve which has large integral points. 相似文献
15.
J. Feuer 《Applicable analysis》2013,92(6):599-606
Our goal in this article is to complete the study of the behavior of solutions of the equation in the title when the parameter p is positive and the initial conditions are arbitrary positive numbers. Our main focus is the case 0 < p < 1. We will show that in this case, all solutions which do not monotonically converge to the equilibrium have a subsequence which converges to p and a subsequence which diverges to infinity. For the sake of completeness, we will also present the results (which were previously known) with alternative proofs for the case p = 1 and the case p > 1. 相似文献
16.
Let k be a field of characteristic ≠ 2 and let Q n,m (x 1, ..., x n , y 1, ..., y m ) = x 1 2 +...+x n 2 ? (y 1 2 +...+y m 2 ) be a quadratic form over k. Let R(Q n,m ) = R n,m = k[x 1, ..., x n , y 1, ..., y m ]/(Q n,m ? 1). In this note we will calculate $\tilde K_0 \left( {R_{n,m} } \right)$ for every n,m ≥ 0. We will also calculate CH 0(R n,m ) and the Euler class group of R n,m when k = ?. 相似文献
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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). 相似文献
19.
We present some comments on the behavior of solutions of the difference equation
where p
i 0, i = 1,..., k, k N, and x
–k
,..., x
–1 R. 相似文献
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