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.
Some Ryle essay footnotes appear on this site, which is devoted to life after physics.
There is no claim of expertise with respect to the musings on this page. Never use my stuff for homework!
Friday, July 14, 2023
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