On 19/05/2013 22:09, Tom Del Rosso wrote:
Martin Brown wrote:
On 19/05/2013 13:15, Jasen Betts wrote:
On 2013-05-19, Martin Brown
wrote:
On 18/05/2013 23:10, Tom Del Rosso wrote:
Martin Brown wrote:

given a^2 + b^2 = c^2 where a,b,c are integers

can you construct A^2 + B^2 = 2c^2 and state A, B

A=B=0

But I get the impression that's not what you had in mind?

I forgot to exclude a=b=0 in the problem statement
although we normally allowed that option as a first try

But there is another solution with a, b, A, B != 0

probably you should have said positive integers at the start.

I should perhaps have said non-zero but it was normal not to bother.

The even powers mean that positive or negative makes no difference.

Anyone that didn't spot a=b=A=B=0 as a solution would fail
immediately.
anyway:

a,b,A,B

5,5 (and more variations on Pythogaras triangle hypotenuse)

1,7 and probably more like that too, certainly multiples of it.

Valid concrete examples I grant you.

The final problem statement is for given a,b,c with

a^2 + b^2 = c^2

find algebraic expressions for A,B

s.t A^2 + B^2 = 2c^2

I wasn't sure how you wanted to define the relationship between a,b and A,B,
but if that is the only thing that relates them, and you just want A,B in
terms of a,b, then...

A^2 + B^2 = 2(a^2 + b^2)

...and simplify, solving for A and B.

You have one equation and two unknowns. How do you propose to do that?

Am I misunderstanding the question again, or are you really just asking for
a restatement of that equation with A and B isolated?

The algebraic solution is in the form

A = F(a,b)
B = G(a,b)

Where A,B,a,b are all integers with some of them non-zero.
In the general case they are all non-zero.
Your task is to find the functions F(), G().

Maybe I forgot how to read math books, but I used to understand the
questions at least, a long time ago.

There isn't a lot of math involved. This is more of a puzzle.
Only basic high school algebra is required.

--
Regards,
Martin Brown