1.Introduction


By Tito Piezas[1],it seems that X^4+Y^4-(Z^4+T^4) = N has a follwing parametric solution.

R. Norrie's Theorem: gAny rational number N is the sum and difference of four rational fourth powers
in an infinite number of ways.h  

((2N+b)c^3d)^4 + (2Nc^4-bd^4)^4 - (2Nc^4+bd^4)^4 - ((2N-b)c^3d)^4 = N(2bcd)^4,  where b = c^8-d^8.

I found a similar parametric solution of X^4+Y^4-(Z^4+T^4) = N.



[1].Tito Piezas:http://sites.google.com/site/tpiezas/001b





2.Theorem
         
     
@@Any rational number N is the sum and difference of four rational fourth powers
    in an infinite number of ways.

    (a(-N+8c^4a^8-8c^12))^4 + (-c(N+8a^12-8a^4c^8))^4 
  - (-a(N+8c^4a^8-8c^12))^4 - (c(-N+8a^12-8a^4c^8))^4 = N(8ac(a^8-c^8))^4.


     
Proof.

X^4+Y^4-(Z^4+T^4) = N....................................................(1)

X=ax+b, Y=cx+d, Z=ax-b, T=cx-d...........................................(2)

Substitute (2) to (1), and simplifying (1),we obtain

(8dc^3+8ba^3)x^3+(8d^3c+8b^3a)x=N


Decide b and d to 8dc^3+8ba^3=0,then

b=c^3,d=-a^3,then

x= -N/8/(ac(a^8-c^8))

Substitute b , d and x to (2),and obtain a parametric solution.            


   
Q.E.D.@
 
@@                  
       
3.Example


N=1..10
(a,c)

(2,1) ( 2039/2040)^4 + (32641/4080)^4 - ( 2041/2040)^4 - (32639/4080)^4 =  1
(2,1) ( 1019/1020)^4 + (16321/2040)^4 - ( 1021/1020)^4 - (16319/2040)^4 =  2
(2,1) (   679/680)^4 + (10881/1360)^4 - (   681/680)^4 - (10879/1360)^4 =  3
(2,1) (   509/510)^4 + ( 8161/1020)^4 - (   511/510)^4 - ( 8159/1020)^4 =  4
(2,1) (   407/408)^4 + (  6529/816)^4 - (   409/408)^4 - (  6527/816)^4 =  5
(2,1) (   339/340)^4 + (  5441/680)^4 - (   341/340)^4 - (  5439/680)^4 =  6
(2,1) ( 2033/2040)^4 + (32647/4080)^4 - ( 2047/2040)^4 - (32633/4080)^4 =  7
(2,1) (   254/255)^4 + (  4081/510)^4 - (   256/255)^4 - (  4079/510)^4 =  8
(2,1) (   677/680)^4 + (10883/1360)^4 - (   683/680)^4 - (10877/1360)^4 =  9
(2,1) (   203/204)^4 + (  3265/408)^4 - (   205/204)^4 - (  3263/408)^4 = 10




 














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