5 ms·
The Buenos Aires Constant
- madcaptenor 2y agoFor those wondering about the name of the constant - its inventors are from Buenos Aires.
- pilaf 2y agoTo add to that, quoting from the paper [1] (published in 2020): > ACKNOWLEDGMENTS: > Dylan, Juli, Bruno and Massi are a group of 18/19-year-old friends from Buenos Aires, Argentina. The original idea came to Juli while having a shower. Bruno calculated the prime-generating constant, first by brute-force and then by finding its formula. As the investigation continued, Juli and Bruno were joined by Massi and Dylan. Later, the team contacted mathematician James Grime who helped by tidying up some of the proofs and writing this note. So ∞ thanks to James! 1: https://arxiv.org/abs/2010.15882 https://arxiv.org/abs/2010.15882
- kragen 2y agoThanks! I was wondering. I hope you're doing well!
- igleria 2y agoAh... FCEN, my alma mater! I'm not saying it as a graduate but my god FCEN has extremely bright students (not me of course, but I did manage to have a decent experience).
- dysoco 2y agoSame, I actually took Calc II with Juli we started FCEN the same year but he obviously was very bright and haven't seen him since I assume he progressed very quickly.
- twic 2y agoGambiarra matemática!
- adityaathalye 2y agoIDK why, but this factoid blowing my mind right now. One of the ways to broadcast evidence of Math + Computing power could be to calculate this constant to an absurd-enough precision and have SETI (or someone) send the digits out into space. Edit: in the sense that, we know how to encode a large set of primes in a single transcendental number, and by the way, here it is.
- H8crilA 2y agoHere's another way: 0.20305070110130 (...) And now this whole shebang looks much less impressive, doesn't it?
- lmm 2y agoWhat comes after 830890970?
- Dylan16807 2y ago10101030107 I'm not sure why you ask? The above number already showed off the transition between 1 and 2 digit primes. And 0s inside the primes should not be a problem.
- tlb 2y agoHmmm, what's the algorithm to extract 101 and not 10101 or 1010103 as the next prime? I suppose you can remember how many digits were in the last prime and only allow the same or 1 more, which should be unambiguous. Though it relies on the unproven conjecture that there are no very long runs (an entire order of magnitude!) with no primes.
- SamBam 2y agoThe point is, with a constructed transcendent number, you can represent any series at all, so long as you have "enough precision." That last part means you can simply adjust your precision to your desired constraints. If you want to unambiguously represent the primes up through all the four-digit primes, simply change your padding accordingly: 0.000200030005.... Obviously the Buenos Aires Constant in the article is a neater representation, and relies on neater math. But GP's point was that you can stuff anything at all into a transcendent number.
- ks2048 2y agoI often prefer code over math notation, but in this case, I find the math statement clearer - see Theorem 1 right at the start of this paper: https://arxiv.org/pdf/2010.15882 https://arxiv.org/pdf/2010.15882 The paper says the mathematician James Grime helped "tiny up the proofs" - some people may recognize him from Numberphile on Youtube.
- dmurray 2y agoThis seems like an incredible magic trick, but once you look closely, it is a magic trick - any series of numbers can be expressed as a constant with this formula. In a different world, we settled on a system of notation based on continued fractions rather than decimals for writing non-integers. In this world, nobody marvels at the irregularity of pi or e, the fact that they seem to go on for ever without a pattern - both numbers have elegant and regular representations as infinite sums of fractions. In this world, we all found it slightly harder to make change at the grocery store, but perhaps we made up for that by producing a million high-school Ramanujans.
- moralestapia 2y ago>it is a magic trick John D. Cook is an expert at posing as a 5-6 figure consultant by making use of extremely trivial Math. Not a complain, but the opposite, I wish I had it like that!
- ttoinou 2y agoI wish you were right but what we see as something simple is sometimes made simple by huge amount of work. Look at how we learn maths and physics right now, it is much simpler than what they had back then. Centuries of human work / knowledge / capital were compressed into tiny powerful equations / definitions / demonstrations, empowering new humans to look at more complex stuff more easily. And there is a good opportunity right now to simplify this even more
- toth 2y agoThat seems uncharitable. I for one enjoy the tidbits he posts in is blog.
- saagarjha 2y agoI am an expert at posing as a 5-6 figure consultant by making use of extremely trivial computer science. I figure a lot of us here are like that.
- moralestapia 2y ago
- bloak 2y agoThe Python code seems to be a kind of arithmetic decoder with the assumption that each prime is less than twice the previous prime. That seems to be a special case of Bertrand's Postulate, but probably there's a lower bound one could use to get a more efficient encoding: https://en.wikipedia.org/wiki/Prime_gap#Upper_bounds https://en.wikipedia.org/wiki/Prime_gap#Upper_bounds
- bloak 2y agoOf course, another way of making this a more efficient encoding would be to take account of the fact that primes tend to be odd: if p is an odd prime then the next prime is one of the (p - 1)/2 values p + 2, p + 4, ..., 2p - 1.
- perching_aix 2y ago> it breaks down around Douglas Adams’ favorite number That being 42 for those wondering (not a book person), so it's pretty early on.
- cluckindan 2y agoYou may be underestimating the geekiness of the crowd here.
- DeathArrow 2y agoWhat's so amazing? That numbers can be expressed as series? The article doesn't tell us what is this constant useful for.
- aqueueaqueue 2y agoThe decoding algorithm is quite elegant
- cluckindan 2y agoSo, which integers are enclosed within e, pi, sqrt(2)…?
- globular-toast 2y agoSo you can write a program of 8+C bytes that can generate ~42 primes in linear time. Is there a theoretical minimum program size that can generate n primes in O(n) time?
- aqueueaqueue 2y agoYou can store a universe in a single real number.
- psychoslave 2y agoBut you’ll need infinite amount of energy to extract it to its fully uncompressed form.
- bambax 2y ago> basically I got about as many primes out of the computation as I put into it. So what's the point of this then?
- unbalancedevh 2y agoIt's an amusing magic trick, for fun. Trivial to those who understand it, but potentially impressive to those who don't.
- jackhalford 2y agoSeems like the constant is another representation of the set of all prime numbers. I wonder of there is a branch of math that formalizes these different representations, there is the infinite series here that defines the constants, but how does that relate to the set of primes? And what are the other representations?
- jks 2y agoOne such representation is a polynomial with integer coefficients in 26 variables given in https://www.tandfonline.com/doi/abs/10.1080/00029890.1976.11994142 https://www.tandfonline.com/doi/abs/10.1080/00029890.1976.11... which has the curious property that as you substitute nonnegative integers for the variables, the positive values of the polynomial are exactly the set of prime numbers. (The polynomial also yields negative values.) When put like this, it sounds like the polynomial must reveal something deep about the primes... but it's another cool magic trick. The MRDP theorem (famous for solving Hilbert's 10th problem negatively) implies that this kind of multivariate polynomial exists for exactly those sets of natural numbers that are computably enumerable, so the polynomials could be seen as a really esoteric programming language for set-enumeration algorithms. More tricks: https://en.wikipedia.org/wiki/Formula_for_primes https://en.wikipedia.org/wiki/Formula_for_primes
- quuxplusone 2y agoThe Python script from TFA: s = 2.920050977316134 for _ in range(16): i = int(s) print(i) s = i*(1 + s - i) You might ask, "Isn't this just continued fractions?" But it's not, quite. The continued-fraction version of that script would be: s = 2.313036736433583 for _ in range(16): i = int(s) print(i) s = 1 / (s - i) The latter (continued-fraction) version is good for only 8 prime terms before it breaks down at the limits of IEEE double precision. The former (TFA, Buenos-Aires-constant) version is good for 13 prime terms! That is, the Buenos Aires protocol is a noticeably better "compressor" than the continued-fraction protocol, given the fixed bit-budget of IEEE double-precision floating point. If I'm reading it right, the explanation is that whereas the continued-fraction protocol is general enough to compress any positive-integer sequence, the Buenos Aires protocol can compress only any monotonically nondecreasing positive-integer sequence where each term is between 1 and 2 times the previous term. (The sequence of primes is such a sequence.) Greater flexibility means lower efficiency, and vice versa.
- aghilmort 2y agointeresting; the Pi of Gödel numbering if knew primes in advance