[HN Gopher] The Buenos Aires Constant
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The Buenos Aires Constant
Author : zdw
Score : 109 points
Date : 2025-02-18 22:30 UTC (3 days ago)
(HTM) web link (www.johndcook.com)
(TXT) w3m dump (www.johndcook.com)
| madcaptenor wrote:
| For those wondering about the name of the constant - its
| inventors are from Buenos Aires.
| pilaf wrote:
| To 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
| kragen wrote:
| Thanks! I was wondering. I hope you're doing well!
| pilaf wrote:
| I'm doing well! Hope you are too.
|
| I sometimes come across your comments here, but I rarely
| have a chance to reply. I almost write the year as 02020
| above just in case you found it, now I'm sad I didn't.
| kragen wrote:
| That's good to hear!
| igleria wrote:
| Ah... 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 wrote:
| Same, 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 wrote:
| Gambiarra matematica!
| adityaathalye wrote:
| IDK 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 wrote:
| Here's another way:
|
| 0.20305070110130 (...)
|
| And now this whole shebang looks much less impressive, doesn't
| it?
| lmm wrote:
| What comes after 830890970?
| Dylan16807 wrote:
| 10101030107
|
| 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 wrote:
| Hmmm, 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 wrote:
| The 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.
| layer8 wrote:
| The fact that any order of magnitude (base 2) contains at
| least one prime was proven in 1852:
| https://en.wikipedia.org/wiki/Bertrand%27s_postulate
|
| Of course, the aliens listening might not have proven it
| yet.
| Dylan16807 wrote:
| Even if that ambiguity did exist, it would just
| complicate the issue of extracting primes from the
| sequence. It wouldn't complicate recognizing the
| sequence.
| ks2048 wrote:
| I 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
|
| The paper says the mathematician James Grime helped "tiny up the
| proofs" - some people may recognize him from Numberphile on
| Youtube.
| dmurray wrote:
| This 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 wrote:
| >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 wrote:
| I 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 wrote:
| That seems uncharitable. I for one enjoy the tidbits he posts
| in is blog.
| saagarjha wrote:
| I 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 wrote:
| >extremely trivial computer science
|
| Indeed!
| toth wrote:
| > 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.
|
| There are some less widely-known topics in math that seem to
| make some of those that learn them want to "evangelize" about
| them and wish they had a more starring role. Continued
| fractions are one.
|
| Now, don't get me wrong. Continued fractions are very cool,
| some of the associated results are very beautiful. More people
| should know about them. But they never will be a viable
| alternative to decimals. Computation is too hard with them for
| one.
|
| Also, while _e_ has a nice regular continued fraction expansion
| [1], that is not the case for _pi_ [2]. There is no known
| formula for the terms, they are as irregular as the decimal
| digits. There _are_ nice simple formulas for _pi_ as _infinite
| sums of fractions_ (simplest is probably [3]) but those are
| _not_ continued fractions.
|
| [1] https://oeis.org/A003417 [2] https://oeis.org/A001203 [3]
| https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
| dmurray wrote:
| > But they never will be a viable alternative to decimals.
| Computation is too hard with them for one.
|
| I think this is too narrow-minded.
|
| You could make the same argument for ideograms vs alphabetic
| writing: one is clearly superior and you could never have a
| technological superpower that relies primarily on the other,
| but thanks to historical path dependency we actually have
| both.
|
| I could imagine a world where the SI system never took off in
| engineering, due to stubborn people at inopportune moments.
| Engineers and physicists would still get their jobs done in
| imperial units, just like American carpenters do today.
|
| Also I did elide the distinction between continued fractions
| and infinite sums of fractions, but again we can use our
| imagination and say that if continued fractions were
| commonplace, we'd all be a lot more familiar with the
| infinite sums too.
| Someone wrote:
| > any series of numbers can be expressed as a constant with
| this formula.
|
| Not _any_ series for this specific approach. As simple
| examples:
|
| - series containing zeroes will lead to division by zero when
| computing the number
|
| - if a series repeats a number, the term for it will be zero,
| and the number cannot be recovered
|
| - if a series grows too fast, the next term 'overflows' into
| the previous one.
| dmurray wrote:
| Good points! Showing what constraints have to be met and
| proving that the prime numbers meet the requirements does
| seem worthy of a paper. But once that paper is in, hopefully
| we don't need 300,000 papers, one for each other sequence in
| the OEIS, proving that it can or can't be expressed like
| this.
| wslh wrote:
| I think another interesting avenue for future research is
| proving whether measures applied to prime numbers (e.g.
| their statistical distribution) have an equivalence in the
| Buenos Aires Constant. Intuitively, it seems likely, but I
| don't have a proof, just a hunch.
| bloak wrote:
| The 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
| bloak wrote:
| Of 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 wrote:
| > 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 wrote:
| You may be underestimating the geekiness of the crowd here.
| DeathArrow wrote:
| What's so amazing? That numbers can be expressed as series?
|
| The article doesn't tell us what is this constant useful for.
| aqueueaqueue wrote:
| The decoding algorithm is quite elegant
| cluckindan wrote:
| So, which integers are enclosed within e, pi, sqrt(2)...?
| globular-toast wrote:
| So 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 wrote:
| You can store a universe in a single real number.
| psychoslave wrote:
| But you'll need infinite amount of energy to extract it to its
| fully uncompressed form.
| bambax wrote:
| > _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 wrote:
| It's an amusing magic trick, for fun. Trivial to those who
| understand it, but potentially impressive to those who don't.
| jackhalford wrote:
| Seems 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 wrote:
| One such representation is a polynomial with integer
| coefficients in 26 variables given in
|
| 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
| quuxplusone wrote:
| The 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 wrote:
| interesting; the Pi of Godel numbering if knew primes in advance
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