Science Forums 2

[CraigD]

Originally Posted by Don Blazys

Now, if you do a "Google search" on "Diophantine equations fourth powers", then you will find the solution that I posted, and many other solutions as well!

Indeed!

From such a search hit, Diophantine Equation--4th Powers -- from Wolfram MathWorld, comes:

Let the notation p.m.n stand for the equation consisting of a sum of m pth powers being equal to a sum of n pth powers. In 1772, Euler proposed that the 4.1.3 equation



had no solutions in integers (Lander et al. 1967). This assertion is known as the Euler quartic conjecture.

So there’s a recognized shorthand notation, “p.m.n”, for equations like

(which would be n.1.m or n.m.1 in the shorthand), and 135 years after Fermat, 237 years before this thread, Euler at least conjectured that the answer to one of post #1’s questions

Originally Posted by CraigD

3. Is there any such that and ?

is “no” for the special case of n.1.m, 4.1.3.

From the same mathworld article:

However, the Euler quartic conjecture was disproved in 1987 by N. Elkies, who, using a geometric construction, found



and showed that infinitely many solutions existed (Guy 1994, p. 140). In 1988, Roger Frye found



and proved that there are no solutions in smaller integers (Guy 1994, p. 140).

So, in disproving Euler’s quartic conjecture, Elkies answered post #1’s question 3 “yes”, 22 years ago.

My “how did you do that?” question (post #7), however, has merely been deflected from “how did you do that?” to “how did he or she do it?", and the search for an answer to

Originally Posted by CraigD

4. Is there a computationally efficient way to generate examples for a given and ?

remains very much still afoot.

Originally Posted by Turtle

is this on the right track?

Binary Quadratic Forms as Equal Sums of Like Powers

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