Prove that x2 y2 z4 has infinitely many solutions with x y

Prove that x^2 + y^2 = z^4 has infinitely many solutions with (x, y, z) = 1.

Solution

Let us substitute z²= w

Then we have x²+y²=w²

where x,y and w are natural numbers can be taken from the Pythagorean triplets

For example, (3,4,5 ) (5,12,13) (7,24,25)(8,15,17)(9,40,41)

and infinitely more

Hence proved that the given equation has infinitely many solutions

(11,60,61)	(12,35,37)	(13,84,85)	(15,112,113)	(16,63,65)
(17,144,145)	(19,180,181)	(20,21,29)	(20,99,101)	(21,220,221)
(23,264,265)	(24,143,145)	(25,312,313)	(27,364,365)	(28,45,53)
(28,195,197)	(29,420,421)	(31,480,481)	(32,255,257)	(33,56,65)
(33,544,545)	(35,612,613)	(36,77,85)	(36,323,325)	(37,684,685)