1.Introduction

Jim Cullen[1] found small solutions of x^4 + y^4 - 1 = z^2 with (x,y)<10000.
According to Tito Piezas[2], E. Fauquembergue gave an identity below.
(17p^2-12pq-13q^2)^4 + (17p^2+12pq-13q^2)^4 = (289p^4+14p^2q^2-239q^4)^2 + (17p^2-q^2)^4.
As Tito points out, integer solution of x^4 + y^4 - 1 = z^2 is derived from the solution of 17p^2-q^2=1.
We searched the identities such as X^4 + Y^4 - Z^2 = W^2 where equation Z = 1 has infinitely many integer solutions.

2. Results

Some identities were found below.
X^4 + Y^4 -Z^2 = W^2 where equation Z = 1 has infinitely many integer solutions.

(4p+3q)^4 +  (2p-7q)^4 -    (4p^2+36pq-39q^2)^2 = (31q^2+8pq+16p^2)^2
(4p+4q)^4 + (2p-15q)^4 -   (4p^2+68pq-191q^2)^2 = (16p^2+120q^2)^2
(4p+5q)^4 + (2p-23q)^4 -  (4p^2+100pq-455q^2)^2 = (16p^2-8pq+271q^2)^2
(4p+18q)^4 + (2p-25q)^4 - (4p^2+156pq-399q^2)^2 = (16p^2+80pq+580q^2)^2

(9p+2q)^4 + (3p-13q)^4 -   (9p^2+84pq-164q^2)^2 = (41q^2+18pq+81p^2)^2
(9p+8q)^4 + (3p-11q)^4 -   (9p^2+96pq-104q^2)^2 = (89q^2+126pq+81p^2)^2
(9p+13q)^4 + (3p-23q)^4 - (9p^2+186pq-479q^2)^2 = (281q^2+198pq+81p^2)^2
(9p+16q)^4 + (3p-22q)^4 - (9p^2+192pq-416q^2)^2 = (356q^2+252pq+81p^2)^2
(9p+39q)^4 + (3p-28q)^4 - (9p^2+318pq-431q^2)^2 = (81p^2+648pq+1656q^2)^2
(9p+45q)^4 + (3p-26q)^4 - (9p^2+330pq-215q^2)^2 = (81p^2+756pq+2124q^2)^2
(9p+59q)^4 + (3p-35q)^4 - (9p^2+438pq-431q^2)^2 = (3665q^2+990pq+81p^2)^2
(9p+73q)^4 + (3p-44q)^4 - (9p^2+546pq-719q^2)^2 = (5624q^2+1224pq+81p^2)^2
(9p+96q)^4 + (3p-50q)^4 - (9p^2+672pq-416q^2)^2 = (81p^2+1620pq+9540q^2)^2


3.Example

(4p+3q)^4+(2p-7q)^4-(4p^2+36pq-39q^2)^2=(31q^2+8pq+16p^2)^2

Intger solutions of 4p^2+36pq-39q^2 = 1 are given by recursion formulae below.

(p(n),q(n))=(1,1).
p(n+1) = 43p(n) + 429q(n).
q(n+1) = 44p(n) + 439q(n).


           3337^4 +            2437^4 - 1 =                      12620311^2 
        1608427^4 +         1174629^4 - 1 =                 2931975736239^2
      775258477^4 +       566168741^4 - 1 =            681162466981740799^2
   373672977487^4 +    272892158533^4 - 1 =      158249026651200010013191^2
180109599890257^4 + 131533454244165^4 - 1 = 36764730368978926259302662855^2

(p(n),q(n))=(-1,-1).
p(n+1) = 439p(n) - 429q(n).
q(n+1) = -44p(n) + 43q(n).


            37^4 +             27^4 - 1 =                      1551^2
         17827^4 +          13019^4 - 1 =                 360175519^2
       8592577^4 +        6275131^4 - 1 =            83676696767719^2
    4141604287^4 +     3024600123^4 - 1 =      19439937546109682151^2
 1996244673757^4 +  1457850984155^4 - 1 = 4516325170503616879761055^2

4.References

[1]:Jim Cullen, x^4 + y^4 - 1 = z^2
[2]:Tito Piezas,x^4 + y^4 - 1 = z^2