Respuesta :
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.
[tex](2^2+1^2)^2=(2^2-1^2)^2+(2\cdot2\cdot1)^2[/tex]
[tex](4+1)^2=(4-1)^2+(4)^2[/tex]
[tex](5)^2=(3)^2+(4)^2[/tex]
So one Pythagorean Triple is (3,4,5).
I'm going to choose (x,y)=(3,1).
x=3 and y=1.
[tex](3^2+1^2)^2=(3^2-1^2)^2+(2\cdot3\cdot1)^2[/tex]
[tex](9+1)^2=(9-1)^2+(6)^2[/tex]
[tex](10)^2=(8)^2+(6)^2[/tex]
So another Pythagorean Triple is (6,8,10).
I'm going to choose (x,y)=(3,2).
x=3 and y=2.
[tex](3^2+2^2)^2=(3^2-2^2)^2+(2\cdot3\cdot2)^2[/tex]
[tex](9+4)^2=(9-4)^2+(12)^2[/tex]
[tex](13)^2=(5)^2+(12)^2[/tex]
So another is (5,12,13).
I'm going to choose (x,y)=(4,1).
[tex](4^2+1^2)^2=(4^2-1^2)^2+(2\cdot4\cdot1)^2[/tex]
[tex](16+1)^2=(16-1)^2+(8)^2[/tex]
[tex](17)^2=(15)^2+(8)^2[/tex]
Another is (8,15,17).
I'm going to choose (x,y)=(4,2).
[tex](4^2+2^2)^2=(4^2-2^2)^2+(2\cdot4\cdot2)^2[/tex]
[tex](16+4)^2=(16-4)^2+(16)^2[/tex]
[tex](20)^2=(12)^2+(16)^2[/tex]
We have another which is (12,16,20).
I'm going to choose (x,y)=(4,3).
[tex](4^2+3^2)^2=(4^2-3^2)^2+(2\cdot4\cdot3)^2[/tex]
[tex](16+9)^2=(16-9)^2+(24)^2[/tex]
[tex](25)^2=(7)^2+(24)^2[/tex]
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).
[tex](5^2+1^2)^2=(5^2-1^2)^2+(2\cdot5\cdot1)^2[/tex]
[tex](25+1)^2=(25-1)^2+(10)^2[/tex]
[tex](26)^2=(24)^2+(10)^2[/tex]
So (10,24,26) is another.
Let (x,y)=(5,2).
[tex](5^2+2^2)^2=(5^2-2^2)^2+(2\cdot5\cdot2)^2[/tex]
[tex](25+4)^2=(25-4)^2+(20)^2[/tex]
[tex](29)^2=(21)^2+(20)^2[/tex]
So another Pythagorean Triple is (20,21,29).
Choose (x,y)=(5,3).
[tex](5^2+3^2)^2=(5^2-3^2)^2+(2\cdot5\cdot3)^2[/tex]
[tex](25+9)^2=(25-9)^2+(30)^2[/tex]
[tex](34)^2=(16)^2+(30)^2[/tex]
Another Pythagorean Triple is (16,30,34).
Let (x,y)=(5,4)
[tex](5^2+4^2)^2=(5^2-4^2)^2+(2\cdot5\cdot4)^2[/tex]
[tex](25+16)^2=(25-16)^2+(40)^2[/tex]
[tex](41)^2=(9)^2+(40)^2[/tex]
Another is (9,40,41).
