1.Introduction

Last time we showed the parametric solutions of X^4 + Y^4 + Z^4 + W^4 = T^2. 
This time we show more simple parametric solutions of X^4 + Y^4 + Z^4 + W^4 = T^2.


2.Theorem
      
 
There are many parametric solutions of X^4 + Y^4 + Z^4 + W^4 = T^2.

     
Proof.

X^4+Y^4+Z^4+W^4 = T^2..................................................(1)

Set X=a, Y=b, Z=a+b....................................................(2)

2(a^2+ab+b^2)^2+W^4=T^2................................................(3)
By eliminating T from equation (3), then we obtain (4).

a^2+ab+b^2 = -2tW^2/(t^2-2), t is arbitrary............................(4)
Transform variables on equation (4), we obtain

A^2+AB+B^2 = -2t/(t^2-2)...............................................(5)

Hence, we should find the rational solution of equation (5).

If we find the rational solution of equation (5), we can obtain the parametric solution of (1).

A necessary and sufficient conditions for solvability of equation (5) is as follows.

-2t(t^2-2) = n^2*p(1)*p(2)*p(3)*....*p(i) where all p(i) are prime and p(i)=1 mod 3.

Some solutions were found; t={-4,-16,-52,-76,-100,-112,-124,-172,-196,...}.

There are many solutions, even only in the case of t = -4, similarly other case of t.

We show the examples for t={-4,-16,-52,-76}.

Q.E.D.


3.Examples
 
(-2+4*k+4*k^2)^4 + (-2-8*k-2*k^2)^4 + (-4-4*k+2*k^2)^4 + (1+k+k^2)^4 = (17*(1+k+k^2)^2)^2

t=-4:
(-6-8*k+2*k^2)^4 + (4-4*k-6*k^2)^4 + (-2-12*k-4*k^2)^4 + (7*(1+k+k^2))^4 = (63*(1+k+k^2)^2)^2

(-4-12*k-2*k^2)^4 + (6+4*k-4*k^2)^4 + (2-8*k-6*k^2)^4 + (7*(1+k+k^2))^4 = (63*(1+k+k^2)^2)^2

(36-4*k-38*k^2)^4 + (2+76*k+36*k^2)^4 + (38+72*k-2*k^2)^4 + (49*(1+k+k^2))^4 = (3087*(1+k+k^2)^2)^2

(2-72*k-38*k^2)^4 + (36+76*k+2*k^2)^4 + (38+4*k-36*k^2)^4 + (49*(1+k+k^2))^4 = (3087*(1+k+k^2)^2)^2

(18-116*k-76*k^2)^4 + (58+152*k+18*k^2)^4 + (76+36*k-58*k^2)^4 + (91*(1+k+k^2))^4 = (10647*(1+k+k^2)^2)^2

t=-16:
(-24+56*k+52*k^2)^4 + (-28-104*k-24*k^2)^4 + (-52-48*k+28*k^2)^4 + (127*(1+k+k^2))^4 = (16383*(1+k+k^2)^2)^2

t=-52:
(-244-564*k-38*k^2)^4 + (282+76*k-244*k^2)^4 + (38-488*k-282*k^2)^4 + (1351*(1+k+k^2))^4 = (1827903*(1+k+k^2)^2)^2

t=-76:
(-488-892*k+42*k^2)^4 + (446-84*k-488*k^2)^4 + (-42-976*k-446*k^2)^4 + (2887*(1+k+k^2))^4 = (8340543*(1+k+k^2)^2)^2


4.Reference
 
[1].Tito Piezas: {16. v^4+x^4+y^4+z^4=nt^k}