Elkies' a^4+b^4+c^4 = d^4

[tpi...@gmail.com]

Hello all,

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

[tpi...@gmail.com]

> Solution v' can be generated by soln v as following
>
> v' = v1/v2
>
> v1 = m6*u^6 + m4*u^4 + m2*u^2 + m0
>
> m6 =(-2364224+6103680*v)
>
> m4 =-16*(190740*v^2-110823*v+56158)*(63580*v^3-110823*v^2+168474*v-57698)
>
> m2 =-8*(63580*v^3-56158*v+28849)*(127160*v^3-110823*v^2+112316*v-28849)^2
>
> m0 = v*(127160*v^3-110823*v^2+112316*v-28849)^4
>
> v2 = n6*u^6 + n4*u^4 + n2*u^2
>
> n6 = -2034560
>
> n4 = 16*(190740*v^2-110823*v+56158)^2
>
> n2 = -8*(190740*v^2-110823*v+56158)*(127160*v^3-110823*v^2+112316*v-28849)^2
>
> n0 = (127160*v^3-110823*v^2+112316*v-28849)^4
>
> v=-31/467
> u=30731278/218089*i
>
> v' =127473934493966820221865642313563283/129759559485872431282952710668698569
>
> Then
>
> 1029222674865727469787858215112904013084138691842718160435340940777832360^4 + 2104913746553430046417859406049840548804152866526338125558359515706490040^4 + 335512010581206396698562077837129837409096932401365003036862114366435489^4
>
> = 2134695166398256785544961408796272986906891690205937773262396417072265377^4- Hide quoted text -
>
> - Show quoted text -

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