[HN Gopher] AI maths whiz creates tough new problems for humans ...
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AI maths whiz creates tough new problems for humans to solve
Author : tzury
Score : 131 points
Date : 2021-02-06 06:37 UTC (16 hours ago)
(HTM) web link (www.nature.com)
(TXT) w3m dump (www.nature.com)
| [deleted]
| David147 wrote:
| The two blog posts by Frank Calegari are from July 2019 and
| irrelevant to the paper in Nature (from February 2021). The paper
| went through peer review by mathematicians and experts in the
| field.
| alisonkisk wrote:
| What was changed in the new paper? Hard to say "irrelevant"
| when the blog post directly influenced a rewrite of the paper.
| scihive wrote:
| There are several new conjectures related to the Catalan
| constant, pi^2, and zeta(3) (Apery's constant):
|
| http://www.ramanujanmachine.com/wp-
| content/uploads/2020/06/c...
|
| http://www.ramanujanmachine.com/wp-
| content/uploads/2020/06/p...
|
| http://www.ramanujanmachine.com/wp-
| content/uploads/2020/06/z...
|
| From looking at Calegari's blog post, he didn't comment on
| any them so far...
| [deleted]
| bluepoint wrote:
| To be fair creating mathematical problems that are hard or
| impossible to solve is not really very hard.
| CyberRabbi wrote:
| > "Until I can detect a well-developed 'sense of mathematical
| taste' in AI, I expect its role to be that of an important
| auxiliary tool, not that of independent discoverer."
|
| AI, no, but ML, yes. ML can only make estimates, e.g. compute
| functions, it doesn't have judgement. That requires a human
| decision maker. It goes back to what I learned in primary school
| about computers: "a computer can only do what you tell it to do."
| magnio wrote:
| The GitHub repo contains more details on how the algorithm works:
| https://github.com/AnonGit90210/RamanujanMachine
|
| > We have continued fractions which are generated by polynomials.
| What does that mean? object of the type:
| a(0)+b(1)/(a(1)+b(2)/...)) where a,b are polynomials with integer
| coefficients. This is the RHS. On the LHS we have a function of
| some constant, either rational or ULCD. A ULCD function is of the
| type f(const)=(u/const + const/l + c)/d while a rational function
| is the ratio of two polynomials. We seek for a match between some
| LHS function on a constant to a continued fraction. To do this,
| we enumerate over the coefficients of a,b and those of the
| rational func on the LHS/the ulcd parameters. They're all
| integers. We actually don't compare between the LHS to a
| continued fraction, but to a continued fraction after we applied
| some function to it, e.g. contfrac^2, sqrt(contfrac), 1/contfrac
| etc. In order to enhance the algorithm's complexity, we first
| enumerate over the parameters of the RHS, saving the results to a
| hashtable, then enumerate over the LHS and look for a match in
| the table. This is a TMTO (time-memory tradeoff), it's called
| MITM (meet in the middle), you can look it up if it's not clear.
| We first find matches, then filter redundant results (discussed
| later on), filter only continued fractions that converge fastly,
| find how fast they are converging, and how many iterations we
| should calculate to get the desired precision. We calculate them
| to that precision, and filter again.
|
| This doesn't seems that complicated, but at the same time
| probably gives no mathematical insights as to how to derive those
| formulaes. I can see how this would look pointless to some
| people.
| warent wrote:
| THIS is mind-bendingly awesome. What are the implications of this
| for technological advancement? Will we be getting to a point
| where machines are inventing new things that revolutionize life
| as we know it? I'm thinking about the difference in technology
| from now vs. 1920s. Could we be moving toward another advancement
| leap of such a proportion?
|
| Just consider for a moment how far we've come. In the past, all
| our machines were purely mechanical---steam engines, water
| cranks, etc. Now we're creating machines that discover math.
| Amazing!
| bawolff wrote:
| The singularity is not yet upon us, and we have had machines
| generating math since the 70s with the proof of the 4 colour
| theorem.
| trenchgun wrote:
| It is intellectual scam.
| layoutIfNeeded wrote:
| Yaay! I fucking love science(tm) too!
| dang wrote:
| " _Don 't be snarky._"
|
| https://news.ycombinator.com/newsguidelines.html
| colordrops wrote:
| It does seem like an amazing project, but I'm struggling to see
| how this has any significant implications for technological
| advancement.
| FrozenVoid wrote:
| Its just a continued fraction generator that looks for
| fractions equal to constants. Not an AI, more like search
| function with optimization.
| heinrichf wrote:
| The praise from popular press and the promotion by the authors
| should be put into context of what mathematicians think of it.
|
| Two blog posts by a professor at U. Chicago, qualifying it of
| intellectual fraud:
|
| https://www.galoisrepresentations.com/2019/07/17/the-ramanuj...
|
| https://www.galoisrepresentations.com/2019/07/07/en-passant-...
| QuesnayJr wrote:
| This is shocking stuff. I encourage everyone to read these two
| links.
| 0b01 wrote:
| I agree with the article you linked. Mathematical knowledge is
| about compression. Most if not all of these formulae are just
| specializations of known formulae. So the value of this
| approach is questionable. Generating these forms can possibly
| be done in a much simpler way.
| agumonkey wrote:
| who else holds the compression view ?
| David147 wrote:
| Looking at his post, the main criticism is "that the program
| has not yet generated anything new", but the post does not
| refer to the actual results (like formulas for Catalan's and
| Apery's constants).
| [deleted]
| dealforager wrote:
| Man, this is an example of how difficult it is to know what is
| BS or not if you're not an expert on the subject. On one hand,
| this article was published in Nature, which I thought was
| trustworthy. On the other, there's this comment on a social
| media platform that links to a blog that also seems legit. No
| wonder misinformation spreads so fast. Even after reading both,
| I don't know what to make of it. The reaction and comments here
| just confuse me more.
| omginternets wrote:
| Science/Nature are prestigious, but the quality of their
| articles are often questionable. Part of the problem is the
| short format, which makes it difficult to include a lot of
| context and sanity-checking. Another issue is that they
| prioritize the "sexiness" of the research over pretty much
| everything else.
| throwawaygh wrote:
| I'll never understand why Science/Nature carry any currency
| in CS and Math. TBH I consider them negative signals in
| these fields, and I encourage others to do the same when
| hiring -- the same way that a prestigious newspaper or
| magazine would treat someone with a bunch of (non-news)
| Buzzfeed bylines.
|
| There are some exceptions. E.g., a Science/Nature paper
| summarizing several years worth of papers published in
| "real" venues. Truly novel work that's reported on for the
| first time in Nature/Science is almost universally garbage.
| At least in CS/Math.
| kevinventullo wrote:
| In my experience, at least in pure math, publishing in
| Nature/Science doesn't carry any weight. The most
| prestigious journals for a given subfield usually
| specialize in that subfield (with a name like _Journal of
| Geometric Topology_ ), with a few exceptions like Annals
| and JAMS. Even those are still focused heavily on pure
| math; I can't think of any which are cross-disciplinary
| outside of math.
| kevinventullo wrote:
| FWIW that blog is written by one of the top leading number
| theorists in the US today. Of course, his opinions are his
| and you're free to form your own, but just wanted to clarify
| that the blog is very much legit.
| David147 wrote:
| It seems like Calegari is chasing after PR and may be angry
| at computer scientists getting into his field.
|
| His criticism was discussed and found incorrect by the peer
| review process:
|
| https://static-
| content.springer.com/esm/art%3A10.1038%2Fs415...
| alisonkisk wrote:
| The authors tweeted at Elon Musk and Yuri Milner, so it's
| obvious who is chasing PR (and "dumb money")
|
| Meanwhile, the blog author congratulated Mathematica for
| being for being good at solving continued fractions
|
| I'd ask you where the criticism was "found to be
| incorrect", but I know that's absurd (aka, not even
| wrong), as peer review comments are not in the business
| of "finding criticism to be incorrect".
| wefaewofj wrote:
| Assuming that it's the PI/senior author doing all of this
| shameless promotional work, I feel really REALLY bad for
| the grad student(s) on this paper... what a way to start
| your academic career :(
|
| The paper is actually really nice work, but holy jesus
| someone on that author list is making a complete ass out
| of themselves.
|
| Academia isn't startup world. The community is small,
| people have long memories, and I've rarely seen the
| strategy being deployed here work out. It does work
| sometimes, but more often it backfires. Especially for
| folks who aren't yet on a tenure track.
| spekcular wrote:
| This is an incredibly strange slate of reviewers. Only
| the first seems to really understand the mathematical
| context. It's odd to appeal to the peer review process
| when the "peers" are not suited to complete the review.
|
| I assure you that Calegari knows more about number theory
| than any of those referees, and the reasons why the paper
| is bad are well-explained on his blog (cf. the two links
| above) and by referee #1. Speaking of "peer review," look
| at how all the excellent mathematicians commenting on
| that blog agree with him!
| scihive wrote:
| Calegari gets to cherry-pick comments he approves or
| rejects on his blog, so calling it "peer review" is
| taking the concept out of context =).
| timkam wrote:
| I agree. Without being an expert in the field, the 2nd
| and 3rd reviews "smell funny"; they clearly lack depth,
| are very enthusiastic, and don't seem to make a good and
| comprehensive point as to _why_ this paper is supposed to
| be as great as they claim it is. In a strong community,
| any editor should consider these reviews disappointing,
| and any author should at least have mixed feelings about
| them.
| testfoobar wrote:
| This is why I have empathy for conspiracy believers. From
| their perspective, their understanding of the world is
| accurate.
|
| This is also why I see the inevitable failure of social media
| platforms in regulating truth-vs-non-truth.
| TeMPOraL wrote:
| This phenomenon has a name: epistemic learned helplessness.
|
| https://slatestarcodex.com/2019/06/03/repost-epistemic-
| learn...
| QuesnayJr wrote:
| Nature in particular seems vulnerable to the academic
| equivalent of click-bait articles. I think the top journals
| within a specific field are more reliable.
| PartiallyTyped wrote:
| Nature has published some very questionable papers in AI/ML
| that are filled with malpractices. Another bogus paper that
| comes to mind was predicting earthquakes with a deep(read
| huge) neural network that appears to have information leakage
| and was fuelled with the hype of DL when a simple logistic
| regression (i.e. single neuron) could perform just as well
| [1,2,3].
|
| [1] https://www.reddit.com/r/MachineLearning/comments/c4ylga/
| d_m...
|
| [2] https://www.reddit.com/r/MachineLearning/comments/c8zf14/
| d_w...
|
| [3] https://www.nature.com/articles/s41586-019-1582-8 /
| https://arxiv.org/pdf/1904.01983.pdf
| lumost wrote:
| This is a frighteningly common practice in DL research.
| Baselines are rarely taken with resect to alternate
| techniques, largely due to publication bias.
|
| On one hand papers about DL applications are of interest to
| the DL community, and useful to see if there is promise in
| the technique. On the other hand, they may not be
| particularly useful to industry, or to forwarding broader
| research goals.
| throwawaygh wrote:
| A good rule of thumb is to be slightly more suspicious of
| "DL for X" unless X was part of the AI/ML umbrella in the
| 2000s. If no one was publishing about X in AAAI/NIPS/ICML
| before 2013 or so then there's a pretty good chance that
| "DL for X" is ignoring 30+ years of work on X. This is
| becoming less true if one of the paper's senior author
| comes from the field where "X" is traditionally studied.
|
| Another good rule of thumb is that physicists writing DL
| papers about "DL for X" where X is not physics are
| especially terrible about arrogantly ignoring 30+ years
| of deeply related research. I don't quite understand why,
| but there's an epidemic of physicists dabbling in CS/AI
| and hyping it way the hell up.
| lumost wrote:
| Anecdotally, having come from a physics background myself
| - DL is more similar to the math that physicists are used
| to than traditional ML techniques or even standard comp-
| sci approaches are. In combination with the universal
| approximation proofs of DL, it's easy to get carried away
| and think that DL should be _the_ supervised ML
| technique.
|
| Curiously, having also spent heavy time on traditional
| data-structures and algorithms gave me an appreciation
| for how stupendously inefficient a neural net is and part
| of me cringes whenever I see a one-hot encoding starting
| point...
| throwawaygh wrote:
| Re: similar to the math they know, this makes sense.
|
| I don't understand why over-hyping and over-selling is so
| common with AI/ML/DL work (to be fair, over-hyping is
| more related to AI than physicists in particular. But
| people from non-CS fields get themselves into extra
| trouble perhaps because they don't realize there are old-
| ish subfields dedicated to very similar problems to the
| ones they're working on.)
| PartiallyTyped wrote:
| I think the answer is related to the startup culture, and
| that is to gather funding.
| throwawaygh wrote:
| That's the confusing thing. Appealing to rich twitter
| users/journalists/the public doesn't strike me as a
| particularly good strategy for raising research funds!
|
| Random rich people rarely fund individual researchers.
| More common for them to fund an institute (perhaps even
| by starting a new one). The institute then awards grants
| based on recommendations from a panel of experts. This
| was true before Epstein scandals, and now I cannot
| imagine a decent university signing off on random one-off
| funding streams from rich people.
|
| All gov funding goes through panels of experts.
|
| Listening to random rich people or journalists or the
| public just isn't how those panels of experts work. Over-
| hyping work by eg tweeting at rich/famous people or
| getting a bunch of news articles published is in fact a
| good way to turn off exactly the people who decide which
| scientists get money.
|
| Maybe a particularly clueless/hapless PR person at the
| relevant university (or at Nature) is creating a mess for
| the authors?
| PartiallyTyped wrote:
| While I agree with you, the nature paper that I linked
| above was published by the folks at google of all places.
| I think a valid hypothesis is that work done during
| internships (or even residencies) may not be on-par with
| what NeurIPS/ICLR/etc require but they give publicity and
| thus the PR teams push for that kind of papers.
|
| However, it still does note explain why this kind of
| sloppy work done and published by publicly funded
| research labs, except perhaps as a form of advertisement.
| throwawaygh wrote:
| Well, yeah, corporate research is what it is. A lot of
| the value add is marketing.
| [deleted]
| PartiallyTyped wrote:
| I agree with the sibling comment that whenever ML has not
| been used before on in a field and DL and especially DRL
| (deep reinforcement learning) are used, it is likely that
| the authors are ignoring decades of good research in ML.
|
| After a very theoretical grad course in ML, I have come
| to appreciate other tools that come with many theoretical
| guarantees and even built-in regularization that are less
| Grad Student Descent and more understanding the field.
|
| I think that the hype that was used to gather funding in
| DL is getting projected onto other fields, if only to
| gather more funding.
| bawolff wrote:
| The articles don't contradict each other when it comes to
| cited facts - you can believe both!
|
| I suppose its all in the implications though, which are
| contradicting as the nature article implies it is a big deal.
| The nature article doesn't give any examples of interesting
| conjectures, or examples of interesting consequences if any
| of the conjectures should be true. They talk a lot about
| alternate formulae to calculate things we already know how to
| calculate. Why would we care? Do they have a smaller big-oh?
| Nature references the theory of links between other areas of
| math, if true that's great, but if its true surely they would
| have mentioned an example of such a link? Anyways I lean
| towards this not being that interesting, even if you base
| that just on what the nature article said.
| sanxiyn wrote:
| Re why would we care: this is a search algorithm for
| numerical coincidences. Most numerical coincidences are
| trivial, for example can be derived from hypergeometric
| function relation which was known to Gauss. In fact it
| would be interesting to automatically filter formulae which
| can be derived from hypergeometric function relation... On
| the other hand, numerical coincidences can lead to deep
| theory, monstrous moonshine is a prime example. Hope is
| that by searching for numerical coincidences, we can
| discover one leading to deep theory without already knowing
| that deep theory. This seems reasonable.
| yharris wrote:
| That's a very good point - and it really motivates these
| kinds of computer searches.
|
| The Nature paper has quite a lot of detail in its
| supplementary
|
| https://static-
| content.springer.com/esm/art%3A10.1038%2Fs415...
|
| Table 3 inside also shows new conjectures for constants
| such as Catalan's and zeta(3). These results do not seem
| to trivially arise from known knowledge.
| zests wrote:
| Sometimes popular science is itself the misinformation. The
| authors stretch the findings to land in prestigious journals.
| The news stretches the findings further to sell clicks (c.f.
| Gell-Mann Amnesia effect). The people on the internet
| selectively quote articles and selectively ignore others. The
| algorithm tries to only show you content that you like.
|
| The truth doesn't have a chance.
| catgary wrote:
| Look at the comments:
|
| > The paper is amazingly bad. None of the authors are
| mathematicians as far as I can see. I think the word "new"
| appears 50+ times in the paper. Looks like they updated the
| paper to include your observation from last time about the
| Gauss continued fraction without mentioning the source (the
| authors admit here they read your blog:
| http://www.ramanujanmachine.com/idea/some-well-known-
| results...). Classy!
|
| Just some light plagiarism/academic misconduct!
| Radim wrote:
| Ah yes, the good old "SV culture disrupts X! Revolution at 8
| o'clock!"
|
| There's an arms race:
|
| * People are evolving _memetic resistance_ to the incessant BS,
| ads and bombastic headlines.
|
| * The SV/startup culture is evolving to inject _authenticity_
| to overcome people 's BS defenses, convince them they need a
| change.
|
| Honestly, do you still get excited when you read "AI solves
| X!"?
|
| Probably another huckster peddling empty air, cutting corners,
| externalizing costs. The whole game is tired, and people are
| taking note. Not everything that exists requires a radical
| change.
| mxcrossb wrote:
| > Well ... OK I guess? But, pretty much exactly as pointed out
| last time, not only is the proof only one line, but the nature
| of the proof makes clear exactly how unoriginal this is to
| mathematics
|
| This is what I was wondering about while reading the article.
| If the AI only generates formula for which proofs involve only
| a few trivial steps back to something that is known, then it
| doesn't feel useful. But I feel like the question "what makes a
| good conjecture?" in its own right makes for a very interesting
| discussion.
| bawolff wrote:
| Wouldn't a good conjecture be anything that's interesting if
| true. General bonus points for if intuitively it seems like
| it should be obviously true (or false) but yet is hard to
| prove or if proving it is true would allow you to prove lots
| of other interesting statements.
| alisonkisk wrote:
| Sure. Define "interesting".
|
| What's interesting to me is probably in a standard textbook
| already.
| bawolff wrote:
| If it is in a standard textbook, than its almost
| certainly interesting (although probably not a conjecture
| unless its a pretty advanced textbook)
| prof-dr-ir wrote:
| Mathematical physicist Robbert Dijkgraaf has got you covered:
|
| https://www.quantamagazine.org/the-subtle-art-of-the-
| mathema...
| j7ake wrote:
| I think one interesting lesson from this nice qualification
| here is that at the moment ML methods to learn mathematics may
| look trivial from a professional mathematician (ie the results
| are unoriginal or trivial) but perhaps the target audience of
| this method may be for non professional mathematicians or
| students training to be mathematicians. I could still see this
| ML tool as a way to automate the work of some more "trivial"
| (from the POV of an expert) mathematics, although not the work
| of professional mathematicians.
|
| The knowledge gap in mathematics between professional
| mathematicians and non professionals is vast, and this tool
| could narrow the gap.
|
| I would bet the majority of readers of nature would not be able
| to point out that the outputs of the ML tool were trivial. So
| there is need to narrow this gap.
| alisonkisk wrote:
| Simple results in almost any specialist field would stump
| most readers of Nature. That's not a reason to publish in
| Nature.
| [deleted]
| [deleted]
| boriselec wrote:
| I understand his frustration. But calling it a fraud is a
| little bit too much.
| yharris wrote:
| Full disclosure- I am one of the authors of the paper.
|
| Note that the blog you're citing was written a year and a half
| ago. It refers to a select few conjectures, and naturally has
| no references to the developments in the past year and half
| (which were the main reason the paper got published).
|
| Furthermore, the author of the blog didn't respond to multiple
| emails we sent him, attempting to discuss the actual
| mathematics.
|
| So basically the vast majority of the criticism here, is based
| on a single, outdated blog, by a professor (respected as he may
| be) who has not revisited the issues and new results since
| first posting the blog, and has not given any mathematical
| argument as to why the results shown in the paper (the actual
| updated paper that was published) are supposably unimportant.
|
| Would appreciate your opinions on the matter.
| _8091149529 wrote:
| Not the person you're replying to, but I admit to
| characterizing your paper as "garbage" in another comment
| thread. Since you're inviting discourse, which I greatly
| appreciate, I'm compelled to reply.
|
| 1) To anyone who's studied algebra, it is clear that
| identities of the form LHS = RHS can be obtained by a nested
| application of transformations and substitutions in a
| consistent manner.
|
| 2) Of course, arriving at a new, insightful result often
| involves taking mundane steps. However, in this case, the new
| mathematical discoveries based on the output tableaus of your
| algorithm are hypothetical. Whereas the manuscript (and the
| authors) have already pocketed one of the premium accolades
| in sciences in the form of a Nature publication.
|
| 3) To drive the point above home, do you think the resulting
| mathematical insights themselves, without riding on the "AI"
| novelty aspect, would clear the bar for a Nature (or similar
| high-impact) publication? To be clear, I'm not a
| mathematican, but I believe the answer would be no. Contrast
| this with another AI/ML advance published in Nature quite
| recently: AlphaGo. Note how the gist of their paper,
| superhuman performance in Go, is a self-standing achievement
| that merely makes use of machine learning techniques.
| throwawaygh wrote:
| "garbage" and "fraud" are really strong words.
|
| I would give the actual work behind this paper a "strong
| accept" if the claims were properly scoped, perhaps with a
| weak/borderline score on "significance/impact" since I'm
| not really sure why anyone cares about discovering
| discovering these sorts of identities. Probably a
| Conditional Accept in its current form because of the
| mismatch between actual results + reasonable expectation of
| potential vs. what's claimed.
|
| So, "over-hyped" and "claims wildly out of line with actual
| results" are definitely more than fair statements. "Fraud"
| or "garbage" are way too strong.
|
| Re: Nature, I don't really understand it or care. I can say
| that in my own input to hiring committees I tend to treat
| Nature papers in CS/Math as red flags unless they're
| consolidations of a bunch of other work published in top
| sub-field journals/conferences.
|
| For some reason Nature really loves these "automated
| discovery of random mathematical facts" type of papers. I
| don't understand it. I tend to assume it's click-through-
| rate-driven editorial decision making.
| timkam wrote:
| I think the vast majority of criticism here does not target
| the research per se, but rather the way the results are
| "hyped" and presented as a massive break-through. I agree
| with this criticism, and also think that the two positive
| Nature reviews seem rather shallow, at least from a non-
| expert's perspective (this is not your fault, of course).
| When it comes to long term impact, I'd find it interesting to
| discuss how your work can (ideally) interact with proof
| assistants like Lean. Also, the work around Lean is a good
| example of a "hyped" topic that is presented by its
| contributors with caution and modesty.
| [deleted]
| sradman wrote:
| The Nature article is about [1]:
|
| > The Ramanujan Machine is a computer algorithm, named after the
| Indian mathematician Srinivas Ramanujan, which generates
| conjectures similar to what proposed by Ramanujan.
|
| The #Criticism section states:
|
| > Prof. Frank Calegari criticized the Ramanujan Machine claiming
| that the program has not yet generated anything new and dubbed it
| as "absurd" and an "intellectual fraud".
|
| [1] https://en.wikipedia.org/wiki/Ramanujan_Machine
| David147 wrote:
| Funny that no other name other than Frank Calegari's appear in
| the Wikipedia page
| heinrichf wrote:
| This page appears as if it was written by the authors or the PR
| department of the university...
| tuyguntn wrote:
| no examples of formula's generated by AI and if they are correct
| or proven incorrect? Without examples this seems an advertisement
| to non-existing product. Anyone can create AI then with simple
| function pick_N_random(=, +, -, *, /, pi, e, alpha, tetta,
| 1,2,3,4,5,6,...)
| buryat wrote:
| the link to the project is in the article
| imhoguy wrote:
| AI? Much overused buzzword these days. Don't we have
| metaheuristics (genetic algorithms etc.) which already do
| exactly what the article states as novel?
| batterylow wrote:
| Meta-heuristics aren't the current buzzword, likewise
| evolutionary algorithms won't be front paging anytime soon...
| unless it's neuroevolution disguised as a novel contribution
| in 2021. Sprinkle on some blockchain and you're onto a
| winner!
| jzer0cool wrote:
| TLDR;
|
| AI machine called: Ramanujan Machine.
|
| Attempts at finding formula. Tries to determine whether the
| following constants are "irrational" or "rational" number.
|
| Constants mentioned: Catalan's constant, Apery's constant
|
| Unrelated: I never heard of the above 2 constants before.
| Although, I have met someone who speaks Catalan and from
| Catalonia.
| tecleandor wrote:
| That names comes from the Belgian mathematician Eugene Charles
| Catalan.
| David411 wrote:
| The paper offers new and improved convergence rates for the
| Catalan constant, presenting new results in a short period of
| time.
|
| Calegari talking nonsense..
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(page generated 2021-02-06 23:02 UTC)