Claim: Every odd integer square is an element of some Pythagorean triple.
We first show that every positive odd number 2n+1 is equivalent to the expression (n+1)2 - n2.
We have (n+1)2 - n2 = (n+1+n)(n+1-n) = 2n+1.
An arbitrary odd square m2 = 2n+1 = (n+1)2 - n2.
That is, m2 = (n+1)2 - n2 or
m2 + n2 = (n+1)2.
Note that we may let n+1 = k, for k2 - (k-1)2
= (k+k-1)(k- k- 1) = -(2k -1) = 1 - 2k.
So, expressed this way, we obtain all the negative odd numbers.
We see this by
2n + 1 = 1 - 2k
or 2n = -2k
or n = -k.
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Friday, July 14, 2023
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