For those of us enamored with the subtle obscurities of the fundamentals, much to my delight a recent posting on 𝕏 (Twitter) by Dr. James Tanton was brought to my attention. Here he describes a unique Pythagorean Triple curiosity.
In subsequent correspondence, Dr. Tanton recalled to me his thoughts of Pythagorean Triples with side lengths as powers of two. He stated:
“There is exactly one integer right triangle with a side length of 4, two with a side length of 8, three with a side length of 16, and so on, N-1 with a side length of 2N.”
His thinking evolved into an idea for finding solutions to xa + yb = zc from a given Pythagorean Triple (a,b,c)… absolutely fantastic!
After suppressing my jealousy, I thought I’d take a stab at it myself.
This membership test strategy seems to work well. However, given the size of the values generated, the requisite resolution of my calculator for exacting integers is quickly exceeded. While not the point of this curiosity, the example given forms a degenerate triangle (line). It would be interesting if one could find a triple-triple, i.e., a right triangle of integer side lengths whose “Pythagorean” compliance results from exponents of another Pythagorean Triple. This configuration would likely have excessively on side lengths. Nevertheless, I suspect there are infinitely many Tanton Triples to be found as both a function of Pythagorean Triples and the a:b ratio.
I highly recommend those of affliction as myself follow the postings of Dr. Tanton.
Thinking Mathematics! (jamestanton.com)
Many thanks James, for making my day.
