Posted by:
Brother Of Jerry
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Date: August 05, 2021 02:15PM
Well, will wonders never cease. Didn't expect to see my old bud Pythagoras in an RFM thread.
Poster "anybody" is correct that the Egyptians knew of some Pythagorean triples. The Babylonians went beyond that in that they knew the formulas that will generate as many unique Pythagorean triples as you care to crank out.
The equations are pretty simple. The trick is realizing that there are equations, and then figuring out what they are. That was the clever trick.
This is not a new discovery about Babylonian math. It's been known for over a century, plus it only scratches the surface of what Babylonians had discovered. I've been threatening to write a post about that, and suppose I need to get on the stick before SC steals my thunder, and DR1 attributes all their discoveries to the Anunnaki.
I assume most people think serious mathematics began with Pythagoras. I know I did. Some of y'all probably even know when Pythagoras lived (around 600 BCE).
Euclid ws about 300 years later, and was most famous for writing a treatise on plane geometry (with a few chapters of number theory, like how to calculate a greatest common divisor).
Euclid's book is still in print. Barnes and Noble carries two editions, one plain, one annotated. It is quite the tome, but cheap since there is no copyright to pay. A textbook that is still in print after 2300 years is a record that will almost certainly never be broken. The book was actually still used as a geometry textbook well into the 1800s, until publishers and authors discovered there was money to be made selling spiffed up editions with color pictures and things like that.
BTW, a Pythagorean triple is three integers a, b, and c that have the relationship that a^2 + b^2 = c^2. It is a bit of a surprise that any integers would have such a relationship. If you don't restrict yourself to integers, you can take any two numbers for a and b, and calculate c as long as you know how to do a square root (which is easy. You hit the square root key on your $7 calculator :)
3, 4, and 5 is the smallest Pythagorean triple. 5, 12, 13 is the next one.
An obvious extension question is: are there equivalent Pythagorean triples for powers higher than 2? For example, are there any triples such that a^3 + b^3 = c^3 ?
Turns out the answer is "no", and that that proposition was extremely difficult to prove, though very simple to state. 17th century French attorney who was more famous as an amateur mathematician, Pierre de Fermat, scribbled in a math book that he had found a marvelous proof that the relationship was impossible for powers higher than 2, but the margin of the book was too small to contain it. Some time thereafter Fermat died, and it took another 4 centuries of concerted effort to come up with a very long and complicated proof.
For the details,
https://en.wikipedia.org/wiki/Fermat's_Last_TheoremFermat was almost always right when he said something was true, but did not supply the proof, so there was hope that there actually was a short clever proof, but the general consensus now is that he was wrong about his "last theorem". The theorem was in fact true, but most mathematicians who care believe that there is no simple proof.
But getting back to the original topic, a^n + b^n = c^n has an infinite number of trivial integer solutions for n = 1. 3^1 + 4^1 = 7^1. Yeah, it's just plain old addition. Any two integers can be added together, and the answer will always be another integer.
There are an infinite number of less trivial solutions for n = 2. 3^2 + 4^2 = 5^2. 5^2 + 12^2 = 13^2. And on and on. You can't take any pair of integers for a and b, and find a "c", but there is a formula which will tell you which integer pairs will have a solution, and the Babylonians figured it out.
And there are no integer solutions at all for all values of n greater than 2. It took nearly 4,000 years to add that little bit of knowledge.