Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.
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Tags: calculus, integration, Diophantine equation, Diophantine Equations
TTsphn
22.10.2007 18:30
We has a result in Pell's equation: 1.Consider the equation: $ ax^2 - by^2 = n$ where $ a,b\in N$,$ n$ is diffent from 0 and $ ab$ is not a square . If the equation has a positive solution then it has infinite solution. Consequence: If $ c$ is different from 0,$ a$ is a solution and isn't a perfect square,$ b^2 - 4ac$ different from 0 then the equation $ ax^2 + bx + c = y^2$ if has a positive solution then it has infinite solution .(1) Now consider the $ f(x,y)$ $ f(x,y) = ax^2 + x(by + d) + cy^2 + ey + d$ $ \Delta = (by + d)^2 - 4a(c^2 + ey + f) = (b^2 - 4ac)y^2 + 2y(bd - 2ac) + d^2 - 4af$ We prove that exist infinite $ y$ to $ \Delta = m^2$ Enough to prove it satisfy condition (1) a)Because the equation has a solution then exist an integer $ y$ to $ \Delta$ is a perfect square b)$ b^2 - 4ac > 0$ and it is not an perfect square.(From the condition of this problem). c)$ \Delta_{1}' = - 4a(acf + bde - ae^2 - cd^2 - fb^2)$ is diferent from 0. So it has many integer solution.
Peter
23.10.2007 00:40
What do you mean by '$ a$ is solution'? And how do you transform to a pell equation on that same line? On the $ \Delta$: how do you know $ (b^2-4ac)$ isn't a perfect square? On $ \Delta_1$, can it be you have a typo there? Else I don't understand what you want to say...