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1.
S. Sh. Kozhegel’dinov 《Mathematical Notes》2011,89(3-4):349-360
We study the set of all natural solutions of the equation x 4 + y 2 = z 2, obtain general formulas describing all such solutions, and prove their equivalence. 相似文献
2.
Periodica Mathematica Hungarica - In this paper we find all positive integer solutions (x, y, n, a, b) of the equation in the title for non negative integers a... 相似文献
3.
Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x 2 + (3n 2 + 1) y = (4n 2 + 1) z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n 3 + 3n, 1, 3). 相似文献
4.
Maohua Le 《Czechoslovak Mathematical Journal》2006,56(4):1109-1116
Let a, b, c, r be positive integers such that a
2 + b
2 = c
r
, min(a, b, c, r) > 1, gcd(a, b) = 1, a is even and r is odd. In this paper we prove that if b ≡ 3 (mod 4) and either b or c is an odd prime power, then the equation x
2 + b
y
= c
z
has only the positive integer solution (x, y, z) = (a, 2, r) with min(y, z) > 1. 相似文献
5.
Liqun Tao 《The Ramanujan Journal》2009,19(3):325-338
In this paper we consider the Diophantine equation x
2+5
m
=y
n
, n>2, m>0. We prove that the equation has no positive integer solutions when 2∤
m, nor when 2∣m under the additional condition (x,y)=1, with the help of Bilu, Hanrot, and Voutier’s deep result in (J. Reine Angew. Math. 539:75–122, 2001).
Supported by the 973 Grant of P.R.C and SRFDP 20040284018. 相似文献
6.
尚旭 《纯粹数学与应用数学》2017,33(4)
在高斯整环中,利用代数数论与同余理论的方法,讨论了不定方程x~2+4~n=y~(13)(n=4,5,6)的整数解问题,得出了当n=4,5时无整数解;n=6是仅有整数解(x,y)=(64,2)和(x,y)=(-64,2)的结论,推进了不定方程整数解的研究. 相似文献
7.
Wang Xiaoying 《Periodica Mathematica Hungarica》2013,66(2):193-200
For any fixed positive integer D which is not a square, let (u, υ) = (u 1, υ 1) be the fundamental solution of the Pell equation u 2 ? Dυ 2 = 1. Further let $\mathbb{D}$ be the set of all positive integers D such that D is odd, D is not a square and gcd(D, υ 1) > max(1, √D/8). In this paper we prove that if (x, y, z) is a positive integer solution of the equation x y + y x = z 2 satisfying gcd(x, y) = 1 and xy is odd, then either $x \in \mathbb{D}$ or $y \in \mathbb{D}$ . 相似文献
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Andrew Bremner 《manuscripta mathematica》1989,65(4):479-487
We determine all cubic number fields such that the title equation has a solution in the ring of integers of the field.I am grateful to H. M. Edgar for suggesting this problem. 相似文献
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12.
Periodica Mathematica Hungarica - Let $$a>1,b$$ be two positive integers where the square-free part of b is 2pq with p, q two distinct odd primes. Recently, Cipu (Proc Am Math Soc... 相似文献
13.
14.
Le Maohua 《Monatshefte für Mathematik》1995,120(2):121-125
For anyD
1,D
2, leth(-D
1
D
2) denote the class number of the imaginary quadratic field
. In this paper we prove that the equationD
1
x
2+D
2
m
=4y
n.D
1,D
2,x, y, m, n, gcd (D
1x,D
2y=1,2m,n an odd prime,nh(-D
1
D
2, has only a finite number of solutions (D
1,D
2,x,y,m,n) withn>5. Moreover, the solutions satisfy 4y
n相似文献
15.
本文运用Baker方法证明了:当D=67时,方程x2+D=yn,x,y,n∈N,n>2,仅有解(x,y,n)=(110,23,3);当D=43或163时,该方程无解 相似文献
16.
本文运用Baker方法证明了:当D=67时,方程x2+D=yn,x,y,n∈N,n>2,仅有解(x,y,n)=(110,23,3);当D=43或163时,该方程无解 相似文献
17.
18.
The Ramanujan Journal - Let C and D denote positive integers such that $$CD>1$$ . In this paper we investigate the solvability of the Diophantine equation $$Cx^{2}+D=2y^{q}$$ , in positive... 相似文献
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