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: “Any rational number N is the sum and difference of four rational fourth powers
in an infinite number of ways.”  

((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.

We obtained a similar parametric solution of X^4+Y^4-(Z^4+T^4) = N  before.[2]

    (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.


This time, we generalized above parametric solution a little.



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

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

    X = a(-8a^4b^8c^4-8a^8d^4b^4+3a^8Nb^4+8a^4b^8d^4+3a^4Nb^8+32b^5cd^3a^7
        -48d^2b^6c^2a^6+32db^7c^3a^5-32d^3b^9ca^3+48d^2b^10c^2a^2-32db^11c^3a+8b^12c^4+Nb^12+Na^12)

    Y = b(-8a^4b^8c^4-8a^8d^4b^4+8a^8b^4c^4+3a^8Nb^4+3a^4Nb^8-32bcd^3a^11
        +48b^2c^2d^2a^10-32db^3c^3a^9+32b^5cd^3a^7-48d^2b^6c^2a^6+32db^7c^3a^5+Nb^12+Na^12+8d^4a^12)

    Z = a(8a^4b^8c^4+8a^8d^4b^4+3a^8Nb^4-8a^4b^8d^4+3a^4Nb^8-32b^5cd^3a^7
        +48d^2b^6c^2a^6-32db^7c^3a^5+32d^3b^9ca^3-48d^2b^10c^2a^2+32db^11c^3a-8b^12c^4+Nb^12+Na^12)

    T = -b(-8a^4b^8c^4-8a^8d^4b^4+8a^8b^4c^4-3a^8Nb^4-3a^4Nb^8-32bcd^3a^11
        +48b^2c^2d^2a^10-32db^3c^3a^9+32b^5cd^3a^7-48d^2b^6c^2a^6+32db^7c^3a^5-Nb^12-Na^12+8d^4a^12)

    W = 8ab(a^8-b^8)(da-bc)^3
    
    a,b,c, and d are arbitrary.
     
Proof.

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

X=ax+c, Y=bx+d, Z=ax+e, T=bx+f...........................................(2)

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

(-4ea^3+4db^3-4fb^3+4a^3c)x^3
+(-6f^2b^2-6e^2a^2+6d^2b^2+6a^2c^2)x^2
+(-4f^3b+4c^3a-4e^3a+4d^3b)x
+c^4+d^4-f^4-e^4=N


Decide e and f to {-4ea^3+4db^3-4fb^3+4a^3c=0,-6f^2b^2-6e^2a^2+6d^2b^2+6a^2c^2=0}, then

e = (2db^3a-b^4c+a^4c)/(a^4+b^4), f = -(-2ba^3c-db^4+da^4)/(a^4+b^4), then

x= 1/8(Na^12-8bcd^3a^11+24b^2c^2d^2a^10-24db^3c^3a^9-8a^8d^4b^4
   +3a^8Nb^4+8a^8b^4c^4+32b^5cd^3a^7-48d^2b^6c^2a^6+32db^7c^3a^5
   +3a^4Nb^8+8a^4b^8d^4-8a^4b^8c^4-24d^3b^9ca^3+24d^2b^10c^2a^2-8db^11c^3a+Nb^12)
   /((a^4+b^4)ba(d^3a^7-3bcd^2a^6+3b^2c^2da^5-b^3c^3a^4-d^3b^4a^3+3b^5cd^2a^2-3b^6c^2da+b^7c^3))

Substitute e, f, and x to (2), and obtain a parametric solution.            


   
Q.E.D.　
 
　　                  
       
3.Example


N=1..3
(a,b,c,d)

(2,1,3,1)(      4793/2040)^4 + (      6833/4080)^4 - (      5033/2040)^4 - (      2993/4080)^4 =  1
(3,1,2,1)(     68841/6560)^4 + (    75401/19680)^4 - (     69001/6560)^4 - (    62441/19680)^4 =  1
(3,1,3,2)(   62441/177120)^4 + (  593801/531360)^4 - (   75401/177120)^4 - (  455959/531360)^4 =  1
(3,2,2,1)(  904353/100880)^4 + (  954793/151320)^4 - (  920993/100880)^4 - (  870553/151320)^4 =  1
(3,2,3,1)( 238753/2723760)^4 + (4324393/4085640)^4 - (1586593/2723760)^4 - (2499047/4085640)^4 =  1
(2,1,3,1)(      4853/1020)^4 + (      5873/2040)^4 - (      4973/1020)^4 - (      3953/2040)^4 =  2
(3,1,2,1)(     68881/3280)^4 + (     72161/9840)^4 - (     68961/3280)^4 - (     65681/9840)^4 =  2
(3,1,3,2)(    65681/88560)^4 + (  331361/265680)^4 - (    72161/88560)^4 - (  193519/265680)^4 =  2
(3,2,2,1)(   908513/50440)^4 + (   933733/75660)^4 - (   916833/50440)^4 - (   891613/75660)^4 =  2
(3,2,3,1)( 575713/1361880)^4 + (2618533/2042820)^4 - (1249633/1361880)^4 - ( 793187/2042820)^4 =  2
(2,1,3,1)(       4873/680)^4 + (      5553/1360)^4 - (       4953/680)^4 - (      4273/1360)^4 =  3
(3,1,2,1)(    206683/6560)^4 + (     71081/6560)^4 - (    206843/6560)^4 - (     66761/6560)^4 =  3
(3,1,3,2)(    66761/59040)^4 + (  243881/177120)^4 - (    71081/59040)^4 - (  106039/177120)^4 =  3
(3,2,2,1)( 2729699/100880)^4 + (   926713/50440)^4 - ( 2746339/100880)^4 - (   898633/50440)^4 =  3
(3,2,3,1)(  688033/907920)^4 + (2049913/1361880)^4 - ( 1137313/907920)^4 - ( 224567/1361880)^4 =  3




 

4.References

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

[2]. Any rational number N is the sum and difference of four rational fourth powers
in an infinite number of ways