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March 16th, 2015, 05:40 AM   #1
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Pythagorean Triples

I'm starting my journey of 'number theory' by reading A Friendly Introduction to Number Theory (4th Edition) (Featured Titles for Number Theory): Joseph H. Silverman: 9780321816191: Books .
I'm in the chapter 2 which is Pythagorean Triples. I don't understand. It says there are numbers x,y and z such that $$a=2x+1,$$ $$b=2y+1$$ and $$c=2z.$$My problem is how these equations come?

Last edited by skipjack; April 2nd, 2015 at 12:46 AM.
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March 16th, 2015, 10:26 AM   #2
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It might simply mean that a and b are odd, while c is even.

Last edited by skipjack; April 2nd, 2015 at 12:43 AM.
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April 1st, 2015, 09:45 PM   #3
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121 + 9 = 130 = 81 + 49, what about 130?

Last edited by skipjack; April 2nd, 2015 at 12:40 AM.
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April 2nd, 2015, 12:42 AM   #4
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As 130 is not a perfect square, a Pythagorean triple (a, b, c) cannot have (a, b) = (11, 3) or (9, 7).

The book states "Next, suppose that a and b are both odd, which means that c would have to be even." That's why the book uses integers x, y and z in the way posted above.
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