Toggle #215 78 5️⃣8️⃣6️⃣3️⃣

Yeah. It's nuts. My wife has wanted a Tesla for years. I like them but the price was nuts. Then I checked comparable gas powered cars and the Model Y was shockingly cheaper. That doesn't make it a bargain it is just that every thing is now insane. Next car will be used.

Even that doesn't account for all of it. What you really need to do it add up all the budgets for polities to which you belong to get the true tax burden with inflation included.

WORD5 #656 4/6 ⬛⬛⬛⬛⬛ ⬛⬛🟧⬛⬛ ⬛🟪⬛🟧⬛ 🟪🟪🟪🟪🟪 https://otherstuff.ai/word5/

That is an excellent observation! In a wheel, 1 has to be counted as a prime! One that never gets eliminated from the wheel. It is part of the symmetry. Numbers kind of tumble into p_n# (the product of primes) in the same way they tumble out of 0. Then the pattern repeats. Thus if 1 is coprime to 30 so is 31 and by symmetry, 29. You always get a free twin canidate at the endpoints. It make sense since p_n# is by definition a multiple of everything then p_n#±1 cannot be a multiple. Generally I would define twins as modulo p_n# so 1 and 29 are considered a pair. But that is messy so I just told you about the consequent pair in the lifted set. So yeah there are weird end point considerations.

If you take the first n primes. We'll call n=3. Then sieve the rest of the number line. You get rid of all the compsites of 2, 3, and 5. The very first composite that catches us by surprise isn't 2*7 or 3*7 or even 2*2*2*7, those are already accounted for. No the first unknown composite is 7*7. In this case p_(n+1) = 7. All the things that kinda look like primes below 49 really are prime. So if I have some construction that sieves the first n primes the anything in that construction that is less than p_(n+1)^2 is a prime. My problem is that I need to show that there is not one number but two that have a difference of two. There isn't anything in the definition of primes that insists they be there. I can count the number of twin primes in the whole set and I know the length of the set, so I can calculate the average gap between them directly. That average says that there should be lots of twin pairs less than the square. But the twin primes are under no obligation to be evenly distributed in the first tiny sliver for my benefit. Thus I need to say something else. In this case I am asking "how far apart can successive pairs get?" Empirically it looks like the maximum gap is indeed less than the square of the next prime. So as far as I have results (n=22) I would be guaranteed at least 4 twin primes even if they were as far apart as they could possibly be. The trend is actually increasing! Successive n seem to allow more and more twin primes less than the square. But it is a very close thing. It grows slowly. Also I don't have a formula for it. The trend may reverse after Rayo's number or something. I very much doubt it, we should be able to say something lame like "here is a super slow growing logarithm, that tracks the ratio of p_(n+1)^2/G(n). We don't know if the values are close to the curve but we do know that it crosses it infinitely often"

Lol. I sure haven't. I don't have a couple hours to pour over the construction of the counter example. I just want to trick llms into one upping fable by trying to prove it.

Any fool AI can find a counterexample to the Jacobian Conjecture but a TRULY intelligent LLM would PROVE the Jacobian Conjecture!

Fable is better. But I can only afford a little bit of it a month. Grok is roughly Opus level but I get a gazillion tokens. It writes most programs without error. So good enough for who its for.

I just had Grok port their program to rust and CUDA. Going to see if I push the boundaries of science!