To find a positive whole number point on a hyberbola

Question

In a hyperbola of equation $$ x^2 - y^2 = N $$ where $N$ is a product of two unknown odd numbers, say $x$ and $y$ such that $x > \sqrt{N} > y$ and $x \ne 1$ and $y \ne N$ , there will be a point that will have $x$ and $y$ co-ordinates which are whole numbers. Assuming N has only two factors except 1, is there any way one can find this point or estimate its location?

Answer 1

As hinted by David C. Ullrich's comment:

Suppose $N=ab$, where $a$ and $b$ are odd integers, and suppose $a\le b$. Then $n=\frac{a+b}{2}$ is an integer, as is $m=b-n$. Thus, we have the point $(n,m)$ on our hyperbola, because $$n^2-m^2=(n-m)(n+m)=ab=N.$$

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Upon further discussion, I see that the question was really the following: Suppose $N$ is a large odd number, and we wish to determine if it has factors by examining integer points on the hyperbola $x^2-y^2=N$. We are guaranteed to find four integer points, namely $(\frac{N+1}{2},\pm\frac{N-1}{2})$, and $(-\frac{N+1}{2},\pm\frac{N-1}{2})$. If we find any more, then we have found a factorization: $N=(x-y)(x+y)$.

Thus, the OP wants to know whether there is a geometric strategy for finding such points, and can such a strategy be used as a method for factoring large numbers?

As far as I know, the problem of locating such integer points is no different from the problem of factoring $N$. If $N$ is hard to factor, then the points are hard to find. Factoring has been studied extensively, and I don't know the current best results, but if you want to find out more about this, I recommend asking a new question, and including context about what problem you're really investigating. Failing to include such context can only lead to confusion.