@futurebird
I don't know off hand what it is, but having the same remainder as a prime squared (7²) will have some forced properties.
A delightfully elementary, visual examination of these sorts of things popped up on YouTube recently; they looked at Base or Modulus of 12 as the composite example, and p² mod B is discussed for prime p, B= 7 or 12. https://youtu.be/lJ3CD9M3nEQ?si=iE6x2yHtwoeRp-k5
@futurebird
Refreshed my memory.
24 | p²-1 for all primes p > 3.
Or p²= 24 n + 1 .
So p² mod 60 in {25, 49} .
Whether this connects to Euclid's formula for Pythagorean Triples isn't obvious yet, but that 1², 7² are the predominant units in Ply.322 is suggestive at least on hypoteneuse column. Alas 169-49=120 is not a squared integer m² as desired, so not helping.
@futurebird I think the only composite number<60 AND no common factors with 60
51 is 3x17 (shares 3)
53 is prime, not composite
55 shares 5
57 shares 3 again
59 is prime
So I could imagine that Babylonian/Sumerian mathematicians might think of 49 as the "most prime-y composite, with a secret key associated with the first difficult number number (7)"
Almost half of the numbers on plimpton 322 have a 49 in the 1s place. Maybe it's just because it's a perfect square? The most popular theory is they worked out the triples geometrically to find right angles for surveying. I was toying with having some students translate it and the numbers are much larger than I expected. And it's strange how so many end in 49 or 1 ...
@futurebird we did once read a cool Donald Knuth paper (a computer scientist, not an anthropologist) which took a detailed look at algorithms found on ancient sources and compared them to how modern algorithms are written. depending on how much context exists for this tablet we definitely wouldn't discount the possibility of having a math person take a close look at it to figure out what it's doing.
@irenes
Well, there's a discrepancy between the description of the problem and the examples, but from a number-theoretical viewpoint I'm not sure that it matters @futurebird
I do not get using base 60 when you could have base 64... and think of how crazy the theories would be if the Babylonians had used a base that's a power of 2 and all about computers. LOL.
@tlariv@futurebird no, yeah, we've always thought base 60, or base 12, are both really good choices for a setting where people are going to be doing a lot of integer math. with our modern concepts of decimal points, and calculating devices that give us as much precision as we want, it doesn't really matter as much, but if you want the numbers to work out cleanly you do want a base with lots of distinct prime factors.
@tlariv@futurebird if you haven't noticed, this is a danger you can encounter with mathematicians and math fans where sometimes people will be talking about the best way to do something and we'll start figuring out the worst, for completeness :D
@tlariv@futurebird like if we were saying the BEST way to write times is something specific, and somebody else chimes in saying no the best way is something else, we're having an argument
if we're saying the worst way is this, and somebody else says no wait what about that other thing, we're both making productive contributions that enhance the discussion :D
@futurebird@irenes
I'm experimenting ... generating Pyth. triples via Euclid, I'm getting more 49 unit or remainder than i'd expect from chance, ×2.3 fold.
Not near as strong as you observed in as Plimpton 322, but enough to say there's a something, and that there was a system to P322's algorithm that favored 49 more strongly than Euclid's.
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