1.Introduction

According to Tito Piezas's website[1], diophantine equation x^4 + y^4 - 1 = z^2 has infinitely many integer solutions below.

By using Fauquembergue's identity (17p^2-12pq-13q^2)^4 + (17p^2+12pq-13q^2)^4 - (17p^2-q^2)^4 = (289p^4+14p^2q^2-239q^4)^2 if q^2-17p^2 = ±1.
  
We found other 4 parametric solutions of x^4 + y^4 - 1 = z^2 below, and show it has infinitely many integer solutions.

(2p^2+15pq-4q^2)^4 + (p^2-15pq-8q^2)^4 - (2p^2+9pq+8q^2)^4 = (p^4+66p^3q+17p^2q^2+264pq^3+16q^4)^2

(2p^2+5pq+7q^2)^4 + (4p^2+7pq+5q^2)^4 - (2p^2+5pq+q^2)^4 = (55q^4+94pq^3+95p^2q^2+56p^3q+16p^4)^2

(p^2+7pq+20q^2)^4 + (2p^2+11pq+19q^2)^4 - (p^2+7pq+8q^2)^4 = (535q^4+478pq^3+203p^2q^2+44p^3q+4p^4)^2

(4p^2+19pq+40q^2)^4 + (8p^2+29pq+35q^2)^4 - (4p^2+19pq+13q^2)^4 = (2008q^4+2408pq^3+1455p^2q^2+464p^3q+64p^4)^2

2.Theorem
 
Diophantine equation x^4 + y^4 - 1 = z^2 has infinitely many integer solutions where 2p^2+9pq+8q^2=±1.

x=2p^2+15pq-4q^2
y=p^2-15pq-8q^2
z=p^4+66p^3q+17p^2q^2+264pq^3+16q^4

     
Proof.

Here we treat first identity below.

(2p^2+15pq-4q^2)^4 + (p^2-15pq-8q^2)^4 - (2p^2+9pq+8q^2)^4 = (p^4+66p^3q+17p^2q^2+264pq^3+16q^4)^2..............(1)

Let x=2p^2+15pq-4q^2, y=p^2-15pq-8q^2, z=p^4+66p^3q+17p^2q^2+264pq^3+16q^4.

Hence let 2p^2+9pq+8q^2=±1.....................................................................................(2)

Equation (2) has a solution (p,q)=(1,-1) and D=9^2-64>0, then by Gauss's theorem, we know that equation (2) has infinitely many integer solutions.

Consequently, equation (1) has infinitely many integer solutions.

Similarly, we can say about other identities.

Q.E.D.
　

3.Example

x^4 + y^4 - 1 = z^2 where 2p^2+9pq+8q^2=1 with (p,q)=(1,-1).

[x,y,z]=[1553, 2552,  6944936]
        [6792209, 11141864,  132436310992424]
        [29573306897, 48511703768,  2510640131523668027816]
        [128762171467793, 211219947094472, 47594993329094761637050360616]
        [560630464997494289, 919651601137657784, 902273233367616658143524455962784424]
        [2440984915836918696977, 4004162860133414927528, 17104676998745139105621388965372958884100136]
        [10628047762923479009174033, 17434124173369287456829592, 324258732733788305726932014395972712094239865181096]
        [46274517518783911769025073169, 75908172646687017453621146504, 6147074613676486216516455027030091634006648537678423958824]
        [201479238648737388918856159434257, 330504166269551100623779015079288, 116532023632893519364441690349139444016608357470895068729779801256]
        [877240558802085072568787949151712273, 1439015064029452845428916378034103912, 2209134163064177028082187882153290380155179618758477292141050648181582376]
        


4.Reference

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