# Solve $32x^2 -y^2 = 448$

I am trying to find all integer solutions to the following equation: $$32x^2 - y^2 = 448$$

This is what I have tried so far:

The equation describes a hyperbola, and so I try the usual trick of intersecting the curve with a line of rational slope to find rational solutions first.

Knowing the point (4,8) satisfies the equation, I solve the following system: $$\left\{ \begin{array}{c} 32x^2 - y^2 = 448 \\ y = m(x - 4) + 8 \\ \end{array} \right.$$

After a bunch of algebra, I get: $$x = \frac{-4m^2+16m-128}{32-m^2}$$ $$y = \frac{8m^2-256m+256}{32-m^2}$$

Finally, substituting $m = \frac{u}{v}$, I get: $$x = \frac{-4u^2+16uv-128v^2}{32v^2-u^2}$$ $$y = \frac{8u^2-256uv+256v^2}{32v^2-u^2}$$

Cool, with any choice of $u$ and $v$, I get a rational solution.

But since cancelling the denominators does not work, I do not know how to continue to get integer solutions only.

Is this perhaps not the right way to go? Any help would be much appreciated.

• what can we say about $y^2$ ... – user451844 Oct 4 '17 at 13:08
• This is a generalized Pell equation, so here is a useful link. I am curious if it is possible to use your geometric method to find the integral solutions, though! – André 3000 Oct 4 '17 at 13:12

$y^2$ divisible by $64$.

Let $y=8y_1$.

Thus, we have $$x^2-2y_1^2=14,$$ which says that $x$ divisible by $2$.

Let $x=2x_1$.

Thus, we need to solve $$2x_1^2-y_1^2=7,$$ which reduce to Pell.

https://en.wikipedia.org/wiki/Pell%27s_equation

• How this reduces to Pell, please? – Piquito Oct 4 '17 at 15:16
• @Piquito Use $(x^2-2y^2)(a^2-2b^2)=(ax-2by)^2-2(bx+ay)^2$ and $1^2-2\cdot2^2=-7$. – Michael Rozenberg Oct 4 '17 at 15:27
• Nice. Thanks you very much. – Piquito Oct 4 '17 at 15:30

Four orbits under $$x_{n+2} = 34 x_{n+1} - x_n,$$ $$y_{n+2} = 34 y_{n+1} - y_n.$$

$$(4,8); \; \; (92,520); \; \; (3124,17672); \; \; (106124,600328);$$ $$(8,40); \; \; (256,1448); \; \; (8696,49192); \; \; (295408,1671080);$$ $$(16,88); \; \; (536,3032); \; \; (18208,103000); \; \; (618536,3498968);$$ $$(44,248); \; \; (1492,8440); \; \; (50684,286712); \; \; (1721764,9739768);$$

As sometimes happens, these can be combined into two orbits under $$x_{n+2} = 6 x_{n+1} - x_n,$$ $$y_{n+2} = 6 y_{n+1} - y_n.$$

$$(4,8); \; \; (16,88); \; \; (92,520); \; \; (536,3032); \; \; (3124,17672); \; \;(18208,103000); \; \; (106124,600328); \; \; (618536,3498968);$$ $$(8,40); \; \; (44,248); \; \;(256,1448); \; \;(1492,8440); \; \; (8696,49192); \; \; (50684,286712); \; \; (295408,1671080); \; \;(1721764,9739768);$$

My program calls them $w,v.$

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jagy@phobeusjunior:~$./Pell_Target_Fundamental Automorphism matrix: 17 96 3 17 Automorphism backwards: 17 -96 -3 17 17^2 - 32 3^2 = 1 w^2 - 32 v^2 = -448 Wed Oct 4 07:13:21 PDT 2017 w: 8 v: 4 ratio: 2 SEED KEEP +- w: 40 v: 8 ratio: 5 SEED KEEP +- w: 88 v: 16 ratio: 5.5 SEED BACK ONE STEP -40 , 8 w: 248 v: 44 ratio: 5.63636 SEED BACK ONE STEP -8 , 4 w: 520 v: 92 ratio: 5.65217 w: 1448 v: 256 ratio: 5.65625 w: 3032 v: 536 ratio: 5.65672 w: 8440 v: 1492 ratio: 5.65684 w: 17672 v: 3124 ratio: 5.65685 w: 49192 v: 8696 ratio: 5.65685 w: 103000 v: 18208 ratio: 5.65685 w: 286712 v: 50684 ratio: 5.65685 w: 600328 v: 106124 ratio: 5.65685 w: 1671080 v: 295408 ratio: 5.65685 w: 3498968 v: 618536 ratio: 5.65685 w: 9739768 v: 1721764 ratio: 5.65685 w: 20393480 v: 3605092 ratio: 5.65685 w: 56767528 v: 10035176 ratio: 5.65685 w: 118861912 v: 21012016 ratio: 5.65685 w: 330865400 v: 58489292 ratio: 5.65685 Wed Oct 4 07:15:22 PDT 2017 w^2 - 32 v^2 = -448 jagy@phobeusjunior:~$


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Note that you can simplify and substitute in the following way: $$\begin{array}{lll} y^2 = 2^5(x^2-14)&&\text{substitute } y = 2^2z \\ z^2 = 2(x^2-14)&&\text{substitute } z = 2a\\ 2a^2 = x^2-14&& \\ 2(a^2+7) = x^2&&\text{substitute } x = 2b\\ a^2+7 = 2b^2&&\text{substitute } a = 2c+1\\ 2(c^2+c + 2) = b^2&&\text{substitute } b = 2d\\ 2d^2 = c^2 + c + 2&&\\ 2(d-1)(d+1) = c(c+1)&&\\ \end{array}$$ so that $y = 16c+8$ and $x=4d$. This might work better for your approach.