A Generalization of Fermat’s Last Theorem

[Turtle]

Originally Posted by Tirtle

is this on the right track?

Binary Quadratic Forms as Equal Sums of Like Powers

Originally Posted by CraigD

Maybe. There are a lot of tracks in sight. Which ones are right … well, that’s the rub.

I know the routine.



So the expression for powers of 4 that I quoted is missing a term it seems, namely another . It should read:

My solution is for x=2 & y=2.

[Turtle]

for & ,

=



How very Pythagorean-ish.

[Turtle]

Originally Posted by Turtle

Binary Quadratic Forms as Equal Sums of Like Powers
Sastry, 1934:

Substituting in this equation u = 2 & v = 2 we get a negative term in the sum:



Edit additions:
u=1 & v=2


u=2 & v=1

[Turtle]

Originally Posted by Turtle

Substituting in this equation u = 2 & v = 2 we get a negative term in the sum:



Edit additions:
u=1 & v=2


u=2 & v=1

For powers of 5 Craig gives this example:

The equation I found from Sastry in 1934, does not generate this example using integers for u & v. I wonder if there exist rational numbers u & v in Sastry's equation that generate Craig's example?

I tried working out u = 2 & v = .5, but I was using the Windows scientific calculator & I think it was clipping off the decimal result past 23 places because the equation was very very close to balanced. Is there a balanced equation here or not?

So thens, beyond powers of 5 there is no known equation for powers of 6. Dare we go down that road? Is it time to add a 3rd variable perhaps??

Dat's dat.