Finding parameter solutions for equations

Question

How does one proceed to find parameters for equations with more variables? For example there is $a^2+b^2=c^2$ where a solution would be of form $(a,b,c) = (u^2-t^2,2ut,u^2+t^2)$ as seen on wikipedia. There are more other parametrizations for this. On another post on this site there was a question about $a^2+b^2=2c^2$ and the answer was $(a+b,a-b,c)$ iff a,b,c is pythagorean triple. So i wonder how do i proceed to find such parameters and maybe how do i check that those parameters yield all or just some of the solution for an equation like this: $ab=cd+ef$ over integers

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Oct 08 '19 at 06:54
Check out the articles in this answer here. https://math.stackexchange.com/a/993608/419507.
You can build up the solutions to square - square equations from solving linear diophantine equations. Another way is through stereo- graphic projection, an example of which can be found in the first chapter of the free book you arrive at by googling "The Topology of Numbers" — MaximusFastidiousIrreverence, Oct 08 '19 at 07:10
Here's a link to that book https://pi.math.cornell.edu/~hatcher/TN/TNbook.pdf — MaximusFastidiousIrreverence, Oct 08 '19 at 07:23
https://math.stackexchange.com/questions/2773097/how-to-find-all-rational-solutions-of-x2-3y2-7/2788381#2788381 — individ, Oct 09 '19 at 08:47

score 1 · Accepted Answer · answered Oct 15 '19 at 15:54

1

"OP" inquired about solution for the below equation:

$ab=cd+ef$ ----$(1)$

Above equation $(1)$ has solution given below:

$(a,b,c,d)=[(5k^2-k-2),(4),(7k-6),(2k),(3k-2),(2k+4)]$ --$(2)$

For, $k=7$, we get:

$(a,b,c,e,f)=(236,4,43,14,19,18)$

Note that solution $(2)$ is not a general solution because,

any numerical solution which does not have

a factor of (b=4) on the LHS of equation $(1)$ is

not a solution for $(2)$. For example $(3*7)=(11*1)+(5*2)$,

is not satisfied by $(2)$, but is a solution to equation $(1)$.

answered Oct 15 '19 at 15:54

Sam

26

Hi. Thanks for the answer. May i know how did you find such a solution even if it's not the general one? What steps did you took in order to arive at such solution – aku jack Oct 22 '19 at 16:05
I'm more interested in the process of finding the solution rather the solution itself. Thanks – aku jack Oct 22 '19 at 16:06

Finding parameter solutions for equations

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