what is the speciality?

Question

if a^(2)-b^(2)
=(a-b)^(2)+2b(a-b)
what is the speciality in it?
there is a speciality in it which will be a help for solving FLT in an elementary and simple way.

A^(n)-B^(n)=C^(2)-D^(2)
7^(3)-4^(3)=140^(2)-139^(2)
X^(n)=Z^(2)-Y^(2)
3^(3)=14^(2)-13^(2)

Answer 1

Rajesh, FLT was solved quite a while ago. Not every easy-to-state problem has an easy-to-understand solution; that's why we work hard in grad school. Follow the link to find something *unsolved* to work on that's worth your time. The prime number ones should be interesting and have accessible statements.

Answer 2

There is no great insight there.
It looks like a high school freshman algebra problem to me.

a^(2)-b^(2)

(a-b)^(2)+2b(a-b) = (a-b)((a-b)+2b) = (a-b)((a+b)

So you have shown a^(2)-b^(2) = (a-b)((a+b).

If FLT could have been solved by simple algebraic manipulation, it would have been solved when the marginal notations were first discovered. All the great mathematicians who have worked on it since then are/were not idiots.

Answer 3

what is the specialty? Take the most complicated expression on the right of the equal sign (a-b)^2 and multiply it out to eleiminate the parentheses. Do the same with the next most complicated expression 2b(a-b). Then add like terms and see how they compare to the left of the equal sign.

Answer 4

alot of people tried to solve FLT.
if it could be solved in a elementary and simple way by simple algebraic substitutions it would have been solved ages ago.

Do you identify yourself with Ramanujan ?

dont waste you time with trivialities find a girl.

Answer 5

(a-b)^2 + 2b(a-b)
= a^2+b^2-2ab+2ab-2b^2
= a^2+b^2-2b^2
= a^2-b^2 .......................... (hence proved)