It’s the same old tree as before, pretty much, but hey…
When m is any odd positive integer, choosing a=1 and b=m gives a factorization of m into ab. In this case, the perfect squares of the appropriate range whose difference gives the equation m=ab come from observing that solving the system of equations s-t=1 and s+t=m produces s=(m+1)/2 and t=(m-1)/2. And (s-t)(s+t)=s2-t2, showing s and t to be the perfect squares whose difference is m.
When nontrivial a and b are not known ahead of time – i.e., when one does not already know m’s factorization above and beyond m=1*m – is there some way that m’s numeric relationships with the perfect squares near it, in particular those above it (the ceiling squares, as I have called them, and/or those beyond the least square greater than m) will give clues beyond what I have discovered to facilitate m’s factorization?
This is repetitious of earlier results I have shared, I know. But my morning inspirations that get me up very early, quite frequently, to work on visualizations and implications of these ideas still seem to me to be tugging at a “yes” answer to the question above, one which might someday help me discover what nobody has before. At the very least, it has already provided, and I hope it will continue to provide, fruitful revelations for my own mathematical satisfaction.
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