Answer:
(3,4,5)
(6,8,10)
(5,12,13)
(8,15,17)
(12,16,20)
(7,24,25)
(10,24,26)
(20,21,29)
(16,30,34)
(9,40,41)
Just choose 2 numbers from {1,2,3,4,5,6,7,8,...} and make sure the one you input for x is larger.
Post the three in the comments and I will check them for you.
Step-by-step explanation:
We need to choose 2 positive integers for x and y where x>y.
Positive integers are {1,2,3,4,5,6,7,.....}.
I'm going to start with (x,y)=(2,1).
x=2 and y=1.
![(2^2+1^2)^2=(2^2-1^2)^2+(2\cdot2\cdot1)^2](https://tex.z-dn.net/?f=%282%5E2%2B1%5E2%29%5E2%3D%282%5E2-1%5E2%29%5E2%2B%282%5Ccdot2%5Ccdot1%29%5E2)
![(4+1)^2=(4-1)^2+(4)^2](https://tex.z-dn.net/?f=%284%2B1%29%5E2%3D%284-1%29%5E2%2B%284%29%5E2)
![(5)^2=(3)^2+(4)^2](https://tex.z-dn.net/?f=%285%29%5E2%3D%283%29%5E2%2B%284%29%5E2)
So one Pythagorean Triple is (3,4,5).
I'm going to choose (x,y)=(3,1).
x=3 and y=1.
![(3^2+1^2)^2=(3^2-1^2)^2+(2\cdot3\cdot1)^2](https://tex.z-dn.net/?f=%283%5E2%2B1%5E2%29%5E2%3D%283%5E2-1%5E2%29%5E2%2B%282%5Ccdot3%5Ccdot1%29%5E2)
![(9+1)^2=(9-1)^2+(6)^2](https://tex.z-dn.net/?f=%289%2B1%29%5E2%3D%289-1%29%5E2%2B%286%29%5E2)
![(10)^2=(8)^2+(6)^2](https://tex.z-dn.net/?f=%2810%29%5E2%3D%288%29%5E2%2B%286%29%5E2)
So another Pythagorean Triple is (6,8,10).
I'm going to choose (x,y)=(3,2).
x=3 and y=2.
![(3^2+2^2)^2=(3^2-2^2)^2+(2\cdot3\cdot2)^2](https://tex.z-dn.net/?f=%283%5E2%2B2%5E2%29%5E2%3D%283%5E2-2%5E2%29%5E2%2B%282%5Ccdot3%5Ccdot2%29%5E2)
![(9+4)^2=(9-4)^2+(12)^2](https://tex.z-dn.net/?f=%289%2B4%29%5E2%3D%289-4%29%5E2%2B%2812%29%5E2)
![(13)^2=(5)^2+(12)^2](https://tex.z-dn.net/?f=%2813%29%5E2%3D%285%29%5E2%2B%2812%29%5E2)
So another is (5,12,13).
I'm going to choose (x,y)=(4,1).
![(4^2+1^2)^2=(4^2-1^2)^2+(2\cdot4\cdot1)^2](https://tex.z-dn.net/?f=%284%5E2%2B1%5E2%29%5E2%3D%284%5E2-1%5E2%29%5E2%2B%282%5Ccdot4%5Ccdot1%29%5E2)
![(16+1)^2=(16-1)^2+(8)^2](https://tex.z-dn.net/?f=%2816%2B1%29%5E2%3D%2816-1%29%5E2%2B%288%29%5E2)
![(17)^2=(15)^2+(8)^2](https://tex.z-dn.net/?f=%2817%29%5E2%3D%2815%29%5E2%2B%288%29%5E2)
Another is (8,15,17).
I'm going to choose (x,y)=(4,2).
![(4^2+2^2)^2=(4^2-2^2)^2+(2\cdot4\cdot2)^2](https://tex.z-dn.net/?f=%284%5E2%2B2%5E2%29%5E2%3D%284%5E2-2%5E2%29%5E2%2B%282%5Ccdot4%5Ccdot2%29%5E2)
![(16+4)^2=(16-4)^2+(16)^2](https://tex.z-dn.net/?f=%2816%2B4%29%5E2%3D%2816-4%29%5E2%2B%2816%29%5E2)
![(20)^2=(12)^2+(16)^2](https://tex.z-dn.net/?f=%2820%29%5E2%3D%2812%29%5E2%2B%2816%29%5E2)
We have another which is (12,16,20).
I'm going to choose (x,y)=(4,3).
![(4^2+3^2)^2=(4^2-3^2)^2+(2\cdot4\cdot3)^2](https://tex.z-dn.net/?f=%284%5E2%2B3%5E2%29%5E2%3D%284%5E2-3%5E2%29%5E2%2B%282%5Ccdot4%5Ccdot3%29%5E2)
![(16+9)^2=(16-9)^2+(24)^2](https://tex.z-dn.net/?f=%2816%2B9%29%5E2%3D%2816-9%29%5E2%2B%2824%29%5E2)
![(25)^2=(7)^2+(24)^2](https://tex.z-dn.net/?f=%2825%29%5E2%3D%287%29%5E2%2B%2824%29%5E2)
We have another is (7,24,25).
You are just choosing numbers from the positive integer set {1,2,3,4,... } and making sure the number you plug in for x is higher than the number for y.
I will do one more.
Let's choose (x,y)=(5,1).
![(5^2+1^2)^2=(5^2-1^2)^2+(2\cdot5\cdot1)^2](https://tex.z-dn.net/?f=%285%5E2%2B1%5E2%29%5E2%3D%285%5E2-1%5E2%29%5E2%2B%282%5Ccdot5%5Ccdot1%29%5E2)
![(25+1)^2=(25-1)^2+(10)^2](https://tex.z-dn.net/?f=%2825%2B1%29%5E2%3D%2825-1%29%5E2%2B%2810%29%5E2)
![(26)^2=(24)^2+(10)^2](https://tex.z-dn.net/?f=%2826%29%5E2%3D%2824%29%5E2%2B%2810%29%5E2)
So (10,24,26) is another.
Let (x,y)=(5,2).
![(5^2+2^2)^2=(5^2-2^2)^2+(2\cdot5\cdot2)^2](https://tex.z-dn.net/?f=%285%5E2%2B2%5E2%29%5E2%3D%285%5E2-2%5E2%29%5E2%2B%282%5Ccdot5%5Ccdot2%29%5E2)
![(25+4)^2=(25-4)^2+(20)^2](https://tex.z-dn.net/?f=%2825%2B4%29%5E2%3D%2825-4%29%5E2%2B%2820%29%5E2)
![(29)^2=(21)^2+(20)^2](https://tex.z-dn.net/?f=%2829%29%5E2%3D%2821%29%5E2%2B%2820%29%5E2)
So another Pythagorean Triple is (20,21,29).
Choose (x,y)=(5,3).
![(5^2+3^2)^2=(5^2-3^2)^2+(2\cdot5\cdot3)^2](https://tex.z-dn.net/?f=%285%5E2%2B3%5E2%29%5E2%3D%285%5E2-3%5E2%29%5E2%2B%282%5Ccdot5%5Ccdot3%29%5E2)
![(25+9)^2=(25-9)^2+(30)^2](https://tex.z-dn.net/?f=%2825%2B9%29%5E2%3D%2825-9%29%5E2%2B%2830%29%5E2)
![(34)^2=(16)^2+(30)^2](https://tex.z-dn.net/?f=%2834%29%5E2%3D%2816%29%5E2%2B%2830%29%5E2)
Another Pythagorean Triple is (16,30,34).
Let (x,y)=(5,4)
![(5^2+4^2)^2=(5^2-4^2)^2+(2\cdot5\cdot4)^2](https://tex.z-dn.net/?f=%285%5E2%2B4%5E2%29%5E2%3D%285%5E2-4%5E2%29%5E2%2B%282%5Ccdot5%5Ccdot4%29%5E2)
![(25+16)^2=(25-16)^2+(40)^2](https://tex.z-dn.net/?f=%2825%2B16%29%5E2%3D%2825-16%29%5E2%2B%2840%29%5E2)
![(41)^2=(9)^2+(40)^2](https://tex.z-dn.net/?f=%2841%29%5E2%3D%289%29%5E2%2B%2840%29%5E2)
Another is (9,40,41).