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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
2.
Let \(a,\ b,\ c,\ m\) be positive integers such that \(a+b=c^2, 2\mid a, 2\not \mid c\) and \(m>1\). In this paper we prove that if \(c\mid m \) and \(m>36c^3 \log c\), then the equation \((am^2+1)^x+(bm^2-1)^y=(cm)^z\) has only the positive integer solution \((x,\ y,\ z)\)=\((1,\ 1,\ 2)\).  相似文献
3.
The power graph ΓG of a finite group G is the graph whose vertex set is G, two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of ΓG and characterize the groups achieving the bounds. Moreover, we determine the independence number of ΓG if G is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups G whose power graphs have independence number 3 or n?2, where n is the order of G.  相似文献
4.
令$G$是一个阶为$n$的有限群, $G$上的强幂图定义为: 以$G$为顶点集, 对于两个不同的元素$x$和$y$, 如果存在两个不超过$n$的正整数$n_1, n_2$使得$x^{n_1}=y^{n_2}$, 则$x$和$y$ 之间连一条边. 本文给出了$G$上强幂图的距离矩阵和邻接矩阵的特征多项式, 并且计算了其距离谱和邻接谱.  相似文献
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