Oh! these squares?

Question

In the equation
a^(2) + b^(2) = c^(2)
I can find out the natural solutions of "b" & "c" when "a"(natural) is given.
Is there any method to find out the natural solutions of "a" & "b" when "c"(natural) is given.

Answer 1

Let there be a right-angled triangle.
now,
we know that,
(2n^2+2n+1)=4n^4 + 4n^2 + 1 + 2(4n^3 + 2n+ 2n^2)
=>(2n^2+2n+1)=4n^4 + 4n^2 + 1 + 8n^3 + 4n+ 4n^2
=>(2n^2+2n+1)=(4n^4 + 4n^2 + 8n^3) + (1 + 4n+ 4n^2)
=>(2n^2+2n+1)=(2n^2 + 2n)^2 + (2n+1)^2
This gives solutions
(3,4,5); (5,12,13); (7,24,25); (9,40,41)........................
the above relation is true for all n belonging to natural numbers
this way you can find solution for any pair of sides the 3rd side been given.
put the given side=the one in the equation
find the positive n, solving the quadratic and waala.

Answer 2

Not really. If you think about it, this is the Pythagorean Theorem. You can form a lot (infinite) of right triangles with the same hypotenuse but with different adjacent and opposite side lengths. They would all still fit in the a² + b² = c² formula.

What you are asking is "Can I solve a single equation with two variables." You really need the same number of unique equations as the variables you are trying to solve for.

Answer 3

By 'natural' I assume that you mean positive integers? This is a classic problem. I'd suggest going to a good website and looking it up. I'll see what I can find.

Found it! They're called 'Pythagorean Triples'. Check it out:

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

http://en.wikipedia.org/wiki/Pythagorean_theorem#Pythagorean_triples

Answer 4

Yes use the method of pythogoras theorm or the ascendin descend both will be applicable here.

Answer 5

yes uses the method of the ascending descent and choose the smallest one.