prove it & merry around.?

Question

prove that 1 can only be expressed as difference of two squares nor as two cubes ,two fourth powers etc.
hint :-a^(n)-b^(n)=c^(2)-b^(2)
like 7^(3)-4^(3)=48^(2)-45^(2)
&
x^(n)=y^(2)-z^(2)
like 3^(3)=14^(2)-13^(2).

Answer 1

I'm guessing that you mean squares of integers, because if not, the assertion is false.

But, if you meant integers then it's easy, since Wiles proved Fermats theorem.

Since x^n + y^n = z^n has no solutions (x,y,z) in integers for n>2, let one of the x or y equal 1 (say x) and get

1^n + y^n = z^n

or

1+y^n = z^n => 1 = z^n - y^n

which can have no integer solutions for n>2


Doug

Answer 2

i definitely dont know... sorry...