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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 ${\Gamma }_{G}$ 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 ${\Gamma }_{G}$ and characterize the groups achieving the bounds. Moreover, we determine the independence number of ${\Gamma }_{G}$ 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
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