- #1
nicnicman
- 136
- 0
Hello all,
I'm practicing proofs and I'm stuck. Here it is:
Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x^2 + y^2 = z^2. Evidently I'm supposed to start by setting x, y, and z like this:
x = m^2 - n^2
y = 2mn
z = m^2 + n^2
So then we have:
(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2
m^4 + n ^4 - 2(mn)^2 + 4(mn)^2 = m ^4 + n^4 +2(mn)^2
m ^4 + n^4 +2(mn)^2 = m ^4 + n^4 +2(mn)^2
Now I'm sort of at a standstill. I understand that I can plug any integer into m and n and x^2 + y^2 = z^2 will be true, but I'm not sure how to prove it.
Also, another way to show the proof would be to let x be:
x = 3m
y = 4m
z = 5m
Since any number can be plugged into m then there are infinite solutions.
However, I would like to understand how to derive the proof from my first method.
I'm really trying to understand this stuff so any help would be appreciated.
I'm practicing proofs and I'm stuck. Here it is:
Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x^2 + y^2 = z^2. Evidently I'm supposed to start by setting x, y, and z like this:
x = m^2 - n^2
y = 2mn
z = m^2 + n^2
So then we have:
(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2
m^4 + n ^4 - 2(mn)^2 + 4(mn)^2 = m ^4 + n^4 +2(mn)^2
m ^4 + n^4 +2(mn)^2 = m ^4 + n^4 +2(mn)^2
Now I'm sort of at a standstill. I understand that I can plug any integer into m and n and x^2 + y^2 = z^2 will be true, but I'm not sure how to prove it.
Also, another way to show the proof would be to let x be:
x = 3m
y = 4m
z = 5m
Since any number can be plugged into m then there are infinite solutions.
However, I would like to understand how to derive the proof from my first method.
I'm really trying to understand this stuff so any help would be appreciated.