finding unimodular substitutions

wildrabbitt · 2020-10-06, 13:18

Hi,

does anyone know a method for finding a unimodular substitution from one
binary quadratic form ( \( ax^2 + bxy + cy^2 \) ) to another given that they are equivalent?

I'd like a find a unimodal substitution

\(x\prime = \alpha x + \beta y \)

\(y\prime = \gamma x + \delta y \)

with integer coefficients which transforms

\(29x^2 + 256xy + 565y^2\) into \(x^2 + y^2 \)

Can anyone help?

wblipp · 2020-10-10, 04:54

x' = 5 x + 22 y
y' = 2 x + 9 y

Start with the "obvious" 29 = 5² + 2²

wildrabbitt · 2020-10-10, 16:54

Thanks.

2020-10-06, 13:18	#1
wildrabbitt Jul 2014 3·149 Posts	finding unimodular substitutions Hi, does anyone know a method for finding a unimodular substitution from one binary quadratic form ( \( ax^2 + bxy + cy^2 \) ) to another given that they are equivalent? I'd like a find a unimodal substitution \(x\prime = \alpha x + \beta y \) \(y\prime = \gamma x + \delta y \) with integer coefficients which transforms \(29x^2 + 256xy + 565y^2\) into \(x^2 + y^2 \) Can anyone help? Last fiddled with by wildrabbitt on 2020-10-06 at 13:24 Reason: latex wasn't coming out right

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2020-10-10, 04:54	#2
wblipp "William" May 2003 New Haven 2·7·13² Posts	x' = 5 x + 22 y y' = 2 x + 9 y Start with the "obvious" 29 = 5² + 2²

2020-10-10, 16:54	#3
wildrabbitt Jul 2014 1BF₁₆ Posts	Thanks.