I’ve recently realised something about Pythagorean triads; a topic which very few people I know would be interested in hearing about… so I’m posting in here - a ghost town maths community. (But I’ll also post on mastodon.). Anyway, the realisation is related to complex numbers.
If I have two complex numbers, I can multiply them like this: (x₁+y₁i)(x₂+y₂i), or like this r₁r₂cis(𝜃₁+𝜃₂). So then, if I represent a Pythagorean triad as a complex number, x+yi, with r as the hypotenuse, then multiplying two of these together is guaranteed to produce another triad. The rectangular method of multiplication guarantees integer real and imaginary components, and the polar method guarantees an integer hypotenuse. For example, (3+4i)(3+4i) = -7+24i. And 7²+24²=25².
So that’s a bit interesting. But I have more. Since the polar angle in these triads is always an irrational multiple of 𝜋, repeatedly multiplying by the same triad will never return the angle to where it started. You’ll just get new triads every time. But of course, if we are multiplying different triads together, its easy to come up with different ways of producing the same triad product. Following this line of thinking, we can view the Pythagorean triads as either ‘prime’ or ‘composite’. Any triad can be written uniquely as a product of prime triads - just like with integers. (For this to fully work, we must allow ‘flat’ triads such as (1, 0, 1), (2, 0, 2), etc.)
How can we tell if a triad is prime? Well, I don’t know - other than trying to brute-force the factorisation. If the hypotenuse is a prime number, then the triad is definitely prime. But if it isn’t… I haven’t thought much about that yet, but my current answer is to just check to see if a triad can be made with the factors of the hypotenuse.
Anyway, that’s all I’ve got on that for now. No doubt there’s some fully fleshed out details somewhere on a wikipedia page citing some well known facts from 2000 years ago or whatever. But discovering is more interesting that knowing. So I’m not going to check right now.
Do your “pythagorean” primes line up with gaussian primes? I suspect they would
This was a straight-out-of-bed thought this morning. I have now properly refreshed myself on gaussian primes and they do not line up. Perhaps this is a new classification of complex primes?
Yeah, I’m not really sure. I mean, we’re talking about complex numbers and naming some of them prime - so I suppose in that sense it definitely is a possible classification of complex primes. It may even be related to Gaussian primes. I know for sure that my complex representation of the triads are ‘prime’ if the hypotenuse is prime - and that’s also a condition for a Gaussian prime. What I don’t know is if it is possible to have a composite hypotenuse, and yet still be a prime triad. I haven’ investigated thoroughly enough to work that out.
Why is the polar angle always an irrational multiple of pi?
I claim they must be irrational multiples of pi based on something I learnt from this video (the relevant part is in the last 10 mins).
In the video, Burkard shows a proof that the nice angles 30°, 45°, 60°, etc are the only rational fractions of a circle which also have rational trigonometric ratios. So then, since those angles don’t make Pythagorean triads, I conclude that no triad has an angle that is a rational multiple of pi. (Unless we allow ‘zero’.)