|
|
steiner@xxxxxxxxxxxxx wrote:
> Hello, all!
> It is known that Fermat solved the Diophantine equation x^4 + y ^4
> = z^2 over
> the integers and proved that it has no nontrivial solutions. He used
> the
> method of infinite descent.
> What can one say about the equation x^4 - y^4 = z^^2? Are there any
> nontrivial
> integer solutions? More generally, what is known about the equation
> x^p - y^p = z^2, where p is an odd prime?
If p = 3, x^p +/- y^p = z^2 has solutions
Solutions to x^3 +y^3 +z^3 -3xyz = w^2
are given by
x = a^2 +2bc
y = b^2 +2ac
z = c^2 +2ab
c^2 +2ab =0 gives an infinite no solutions, but not all
I seem to remember the name "Terquem" in relation
to "sums of two equal powers = square", but
this may have been x^2p - y^2p = z^2.
|
|