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The main purpose of this paper is to use the M. Toyoizumi's important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it.

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.

Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E:y^2=x^3-4p^{2}x$. Further, let $N\left(p\right)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\ge 17$, then $N\left(p\right)\le 4$ or $1$ depending on whether $p\equiv 1(mod8)$ or $p\equiv -1(mod8)$.