After some substitution, a particular case of Elkies' solution can be
reduced to the simple identity:
(85v^2+484v-313)^4 + (68v^2-586v+10)^4 + (2u)^4 = (357v^2-204v+363)^4
where,
-22030-28849v+56158v^2-36941v^3+31790v^4 = +/-u^2
This is, of course, an elliptic curve, with one soln (for the - case)
as v_1 = -31/467. From this rational point, one can then find an
infinite number of v_i. (This first value gives, after removing common
factors, Elkies' smallest soln d = 20,615,673.)
P.S. Note that the smallest known solution found by computer
searching, 95800^4 + 217519^4 + 414560^4 = 422481^4, by Roger Frye
belongs to another case of Elkies' solution which gives rise to
another identity.
- Titus
Ah, I know how you found this. But there is a soln with smaller
height:
v_2 = 83713/124659
though, after removing common factors, this gives the same terms as
Elkies' smallest soln.
- Tito