congruent numbers

[gortwell]

congruent numbers [SOLVED]

Hi,

For integers \(\displaystyle a>b\geq c>d>0\)

Is it possible to ascertain whether \(\displaystyle (d^2-c^2)ab\) can produce a congruent number solution apart from by trial and error?

 

[gortwell]

Define congruent.

A positive integer n is said to be congruent if there is a right-angled triangle with rational sides and area n.
This congruent is unrelated to the definition used in modular arithmetic.

i.e. a right-angled triangle with sides 5, 12 and 13 has an area of 30 so 30 is a congruent number

http://en.wikipedia.org/wiki/Congruent_number

The negative value obtained from the given equation is not important in this case as I'm actually interested in | \(\displaystyle (d^2-c^2)ab\) |

 

[guitarguy]

I do not completely see the solution here but I recognize that d^2 - c^2 equals the side squared of a right triangle. If you call the side of right triangle e then e^2 could equal a*b.

I think this would make it possible to be congruent; however, I am not sure.

Interested if anyone has the solution.

 

[gortwell]

thanks for the input.

Found a solution for this part of my problem anyway

The right-angled triangle 8, 15, 17 has an area of 60 therefore 60 is a congruent number

let a=4, b=3, c=3 and d=2

then \(\displaystyle |(d^2-c^2)ab|=60\)

so \(\displaystyle |(d^2-c^2)ab|\) can produce a congruent result