See the below image (I'm not allowed to embedd the image yet). The reading before this exercise in my course gives the proof for the following corollary: Every Pythagorean triple is similar to one arising from the Babylonian method.
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1What have you tried? People here get cranky if you don't show any work/attempts. – scoopfaze Jan 25 '20 at 04:18
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I've been staring at this problem for like 2 hours now. I can't really seem to understand why the order of x,y needs to be changed other than the primitive triple would be in the form (4,3,5) instead of (3,4,5) if we don't change the order. I guess I'm just not really understanding the purpose of the problem. – Frakkin Jan 25 '20 at 04:23
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Better to write out the problem than to link to an image offsite. There's help with formatting math available via the Help menu. – Gerry Myerson Jan 25 '20 at 04:24
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I also found a proof for what I think is the answer in this PDF http://math.uga.edu/~pete/4400triplesnew.pdf, but I also just can't make sense of it. – Frakkin Jan 25 '20 at 04:25
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To Gerry's comment: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – scoopfaze Jan 25 '20 at 04:25
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"I guess I'm just not really understanding the purpose of the problem." The purpose of the problem is to show you that there's a simple formula that gives every solution of $x^2+y^2=z^2$. We don't really want to count $(3,4,5)$ and $(4,3,5)$ as different, so we make the arbitrary decision to make $x$ odd, and $y$ even. – Gerry Myerson Jan 26 '20 at 10:55
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But it's not clear what you want. Do you want a proof of the formula, that is, an answer for the exercise? That has probably been done several times on this website, you could do a search for it. – Gerry Myerson Jan 26 '20 at 10:58