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## Homework Statement

[tex]

\frac{\pi(1+2\alpha)}{t}=x \&\& ((\alpha \geq 0 \&\& 1+ 2\alpha < 4\beta \&\& \pi \sqrt{-(1+2\alpha)^2+16\beta}=2t)\\ ||((1+2\alpha>0 \&\& 2\alpha < 1 +4 \beta \&\& \beta \geq 0 \&\& \pi \sqrt{(-1+2\alpha -4\beta) (3+2 \alpha +4\beta)}=2t)

[/tex]

[tex]

\alpha and \beta[/tex]are integers.

This is a solution I obtained from Mathematica, it's ugly as you can tell. How can I generalize this concisely?

How can I find a general expression for x from this equation?

## Homework Equations

## The Attempt at a Solution

If we look at the first condition, [tex] \alpha \geq 0 [/tex] which is nice

\alpha \beta 1+2*alpha < 4*beta

0 1 1 < 4

1 2 3 < 8

2 3 5 < 12This is false when

\alpha \beta 1+2*alpha < 4*beta

0 0 1 < FALSE

1 1 3 < 4

2 2 5 < 8

Any suggestions?

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