I think I got the idea, I just now updated the original post and added a footnote how I understood the Brahmagupta–Fibonacci identity.I have only surface knowledge about imaginary numbers, but I have noticed sum of two squares being also referred as Gaussian integers.There's definitely a lot of material to read about relating both Gaussian integers and Pythagorean triples, that I can gain intuition how all of these relations work out. I much appreciate the help!
Thank you for the answer! I have to take deeper look into that.It's rather interesting how seemingly two different sum components share factors. It's like finding pi from seemingly unrelated series.
After some thinking I came up with this.Conjecture:
N² = a²+b²
N = c²+d²
a = 2*c*d
b = d²-c²
integers a,b,c,d>0
For any integer N that can be expressed as both N²=a²+b² and N=c²+d², the relationship a=2*c*d holds.
Is it enough proof just to show that the above equations are true when substituting them to N² and N equations?If a = 2*c*d then one leg of the Pythagorean triple definitely contains c and d factors and '2'. This might be related to Pythagorean triple parametrization.
I fixed the flawed ChatGPT counterexample finder script and now it gives this list of all numbers including composites which don't follow these rules, such as 58. Here's sub-sequence of A004431 numbers that don't follow the rules:
Numbers which neither Pythagorean side is divisible by 4Composites: [10, 26, 34, 50, 58, 74, 82, 90, 106, 122, 130, 146, 170, 178...]This is identical to Sums of two distinct odd squares A339977 which makes sense as if c,d>1 and odd, they must be 3 or bigger hence their factors are missing the extra '2', unlike primes where one of c or d is always even.Even composites appear to follow a=2*c*d relation. The prime pythagorean side divisibility by 4 appears to be just a side effect.
I've been using lemmy almost a year now and it has been fairly smooth. But I still get logged out every time I refresh some community pages which is strange.
I agree with the biological definition, "organism that can survive as an individual". Even if the fetus has a parasitic relation, it is capable of developing all functions to fit the full definition.There are other definitions of 'life' and anyone is free to believe either way, but the more subjective question is: When does the fetus become a person?
Same thing with alcohol prohibition. People are going to break the law, legal or not. The US is said to be the freedom land, but women can't have autonomy over their bodies.What I gathered is that 70% of the US congress is men, so it's not their freedom that they sacrifice. Christian values are important to some voters, so politicians can gain free points by promising anti-abortion laws. The politicians who make such decisions think one term at a time and disregard the consequences as long as it doesn't affect them. If they actually cared, they would also advocate for childcare benefits.
I believe abortion is killing, but it only becomes murder by definition if abortion is outlawed. The literal definition is "the unlawful premeditated killing of one human being by another."The fetus won't have cerebral cortex till +12 weeks; no consciousness. So it wouldn't be much different than killing a plant or bacteria. It's debated when it becomes a human.Yes, I justify murder as it will likely reduce the suffering. Thus, I am a monster in Christians eyes.
That was a good watch, I can see binary view being at least part of the problem. I did think that religion might be part of the reason for some laws, but somehow didn't realize how big. And the video had many other good points worth of the watch.
I genuinely thought symlinking these files was standard because how many people I have seen suggesting it. I have had this issue so many times when I needed to that one program updated but there's no newer libraries in AUR. Surprisingly, I haven't had issues and I've been doing this past 5 years on my personal system.
So I guess I consider compiling from source next time.
I decided to spend a day debugging linux boot failure, which I found to be caused by the Nvidia driver.