Re: Quadratic equation



On Fri, 3 Nov 2006, Edwin Klement wrote:

Hello,

can somebody give me a hint how to solve an quadratic equation like 3*x*x
+ y*y - n = 0
where x, y and n are positive integer.

Edwin

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A few starting remarks: I have to guess what you might have meant, namely
"n given, find all pairs of x and y". If that is not the case, please
clarify.

You go by the remainder when n is divided by 3.

Remainder 2: no solution (you can explain why).
Remainder 0: a "descent" is possible, because y must be a multiple of 3,
so
y = 3*z
n = 3*m

and from 3*x^2 + (3*z)^2 = 3*m

you obtain

3*z^2 + x^2 = m

and you repeat until you get a remainder 1 or 2 on the right-hand side.
(Then you find solutions and reverse the substitutions.)

What to do about remainder 1? There is always exhaustive search (not
always practical), and you can get finitely many answers (maybe none at
all). Perhaps someone will pick up from here.

Cheers, ZVK(Slavek).
.

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