一名学生被要求用反证法证明：

“不存在正整数 $x$ 和 $y$ 使得 $3x^2+2xy-y^2=25$”

该学生证明的开头已在下框中给出。

写出完成该证明所需要的计算和论述。

题框已把原式因式分解为 $(3x-y)(x+y)=25$，并处理了因数对 $(1,25)$。因为 $x,y$ 为正整数，所以两个因数都为正；继续检查剩余的正因数对 $(25,1)$ 和 $(5,5)$，证明它们也不能产生正整数解即可。

The case

has already been considered in the question.

Since $x$ and $y$ are positive, $x+y>0$. As

it follows that $3x-y>0$. Therefore, only the positive factor pairs of $25$ need to be considered.

For the factor pair $(25,1)$,

Adding the equations gives

so

These are not positive integers.

For the factor pair $(5,5)$,

Adding the equations gives

so

Again, these are not integers.

The three positive factor pairs of $25$ are $(1,25)$, $(5,5)$ and $(25,1)$, and none gives positive integer values of both $x$ and $y$. This contradicts the original assumption.

Hence no such positive integers $x$ and $y$ exist.