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(Much of this has been heavily edited by Jorend 16:22, 16 Mar 2004 (UTC). See Page history for the original versions, especially if your comment has been deleted.)

Currently this page does contains no discussion of either the Hardy-Ramanujan anecdote or the interesting number paradox. Yet both Hardy-Ramanujan number and interesting number paradox redirect here and are protected.

All known relevant entries:

1729 (number) (this entry)

1729 (anecdote)

Hardy-Ramanujan number

Hardy Ramanujan number

interesting and uninteresting numbers

interesting number paradox

taxicab number

1729 (the year; that entry has a link to this mess)

Jorend 16:45, 16 Mar 2004 (UTC)

• Need to add reference to "Gardner (1): Martin Gardner, Mathematical Puzzles and Diversions, 1959" when this page gets unprotected. [mooted until organisation issue is resolved]

• Should taxicab numbers be capitalised? Anthony DiPierro 18:25, 15 Feb 2004 (UTC) [note that I agree it should not be capitalised - this appears to be resolved]

[No.] Wikipedia talk:Naming conventions (theorems). -- Toby Bartels 06:04, 18 Feb 2004 (UTC)

• Should there be a link to http://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html [this question is mooted until the organisation issue is resolved]

No, a link to the entry Richard's paradox makes sense. -- Jorend 16:48, 16 Mar 2004 (UTC)

• Why is 1729 named Hardy-Ramanujan and not Ramanujan-Hardy? Ramanujan should have top billing - he did the work here. Hardy did nothing other than relate an anecdote. Did Hardy get his name first because he is the white guy?173.76.228.52 (talk) 19:01, 5 March 2018 (UTC)Reply[reply]

[Agree.] --Devinberg (talk) 10:31, 18 February 2020 (UTC)Reply[reply]

Anthony DiPierro added the info about the dullness of 1729 and the external link, and Wik deleted it. When I asked him why he deleted it, he told me that it was solely because Anthony had added it, and he intended to revert every posting Anthony makes. When he refused to stop reverting, I protected the page and banned Wik. Then I unprotected the page. When Secretlondon unbanned Wik, he came back here and reverted the page again, so I rebanned Wik, and reprotected the page. RickK 06:09, 18 Feb 2004 (UTC)

Wik, I agree that Anthony's edit is not very good -- not because I dislike Anthony, but because it has too much material that belongs on the more general page Richard's paradox. So I would like to imporve Anthony's edit, by moving things around and combining material. If I do that, then will you let some material on the paradox remain here, even though Anthony wrote it? If so, then I will edit the page and leave it unprotected. -- Toby Bartels 04:10, 24 Feb 2004 (UTC)

I don't see what of Anthony's edits here is worth keeping, except possibly the external link. Anthony didn't write about the paradox, he denied that it is a paradox. I suggest you revert to my last edit and improve on that. --Wik 05:03, Feb 24, 2004 (UTC)

I'd like to see a respectable source which describes this as a paradox. The sources I've seen, including apparently the original in Mathematical Puzzles and Diversions describes this as a tongue-in-cheek proof. This has very little to do with Richard's paradox. And since the proof is fake (relying on the first uninteresting number being interesting by definition), it's not actually a paradox at all. Why don't you show us your proposed edit here? Anthony DiPierro 15:46, 24 Feb 2004 (UTC)

This source seems to have made a good explanation as to how this can be made into a paradox. Including much of this material would be excellent. We should probably move this whole page to "Interesting and uninteresting numbers" or something like that. Anthony DiPierro 15:57, 24 Feb 2004 (UTC)

A tongue-in-cheek proof by contradiction exists showing that all numbers are "interesting." In a classification of numbers as to whether they had interesting properties or not, there would be a smallest number with no interesting properties (for instance, 38 could be a candidate). This in itself would be an interesting property of the number, making it interesting, and thus excluding it from the list. Therefore, all numbers must be interesting.

This is not a proof, as "this is the 345th smallest uninteresting number" is not a interesting characteristic of a number. —Sverdrup(talk) 19:41, 25 Feb 2004 (UTC)

But "this is the smallest uninteresting number" is an interesting characteristic of a number. So it is a proof, by contradiction. (Of course, such proofs rely on the law of excluded middle, which perhaps is where the whole paradox part comes in.) Anthony DiPierro 20:16, 25 Feb 2004 (UTC)

I agree on the first point, but: it's a ridiculous proof. Say you iterate through all integrers 1-1000 to mark the interesting ones. If 739 of them (for example) are marked interesting, "this is the smallest uninteresting number" and conclusion is no number is uninteresting, is that true? No. I can't possibly see that it's a proof for real. —Sverdrup(talk) 20:25, 25 Feb 2004 (UTC)

Well, it's a tongue-in-cheek proof, because it relies on the smallest uninteresting number being defined as "interesting," but you don't seem to understand the concept of a proof by contradiction. You assume the thing you're trying to disprove, and then you show that it leads to a contradiction. That's how this type of proof works. Anthony DiPierro 21:08, 25 Feb 2004 (UTC)

As a non-native, I'll just accept that tongue-in-cheek-proof means that it's a silly proof, and accept your point. — Sverdrup (talk) 18:39, 3 Mar 2004 (UTC)

(I also removed the tongue-in-cheek link, as Wikipedia is not a dictionary.)

... I thought. But the page seems to be protected :/

Yeah, the page is protected until we come to an agreement as to how to word this. If you have any ideas, please put them below. As for tongue-in-cheek, yes, it means a silly, or fake proof. If you're familiar with the term "sarcastic" (from "sarcasm"), that is a closely related term. Anthony DiPierro 18:44, 3 Mar 2004 (UTC)

Well, I for my part enjoyed the presentation of the tongue-in-cheek proof immensely, it also very nicely fits in with the cited discussion on top of the page (dull number). Very well done, Anthony. That's what you're really good at. --Palapala 21:07, 3 Mar 2004 (UTC)

Oh I wish it were that simple and I could just say "thanks" and be done with this. However, the original version looked like this:

"The issue of classifying numbers as "dull" and "interesting" leads to an interesting paradox (strictly speaking, an antinomy). In a classification of numbers as to whether they had interesting properties or not, there would be a smallest number with no interesting properties (for instance, 38 could be a candidate). This in itself would be an interesting property of the number, making it interesting, thus excluding it from the list. Closely related paradoxes/antinomies are the Berry paradox and the Liar paradox."

The question is, what is it, a paradox, a tongue-in-cheek proof, both, neither? Is it an antinomy, an antinomy of definition? My investigation has led me to believe that this was first presented as a humorous proof made by Martin Gardner in Mathematical Puzzles and Diversions, 1959. However, I've read other pages which suggest that it can be made into a paradox of sorts (noting that a paradox is merely something that seems to lead to a contradiction, not something that actually is a contradiction). Now, I'm perfectly willing to work with someone to try to flesh out how this should be worded, however Wik was reverting every change I made without comment. So the page is currently protected. Anthony DiPierro 21:15, 3 Mar 2004 (UTC)

Hmm. What definition of "paradox" are we using here? Obviously this proof is not a formal logical paradox. However "paradox" can also more loosely mean a logical result that contradicts reasonable intuition (ie, the intuition that some numbers really are uninteresting) -- and in this sense it can be termed a paradox. (I'm not quite sure what an "antinomy" is -- wikipedia defines it as a term which "loosely, means a paradox.") As for whether it's tongue-in-cheek or not, it's certainly not a valid proof. It relies on a contradictory assumption -- that "the smallest uninteresting number" is in the set of interesting numbers. How could a number be both interesting and uninteresting? This flawed definition, and not anything inherent in numbers themselves, is what leads to the strange result. (Another way to word it -- you're saying that the "smallest number with no interesting properties" has an interesting property. But it can't have an interesting property and have no interesting properties. Contradiction.) PenguiN42 15:31, 25 Mar 2004 (UTC)

a) Should the statement that a number is interesting because it is the smallest uninteresting number be treated as an axiom? Acceptance or rejection of the axiom would be a matter of individual choice.

b) As the application of this putative axiom to the smallest uninteresting number leads to its removal from the list of uninteresting numbers and places the next lowest uninteresting number in a similar position that also requires its removal, should this be regarded as proof by mathematical induction rather than proof by contradiction? Xenoglossophobe 01:41 26 Mar 2004 (GMT)

I believe that this page should stay, because I like the Ramanujan story. The wording of the current Interesting and Uninteresting Numbers section is good, and it should stay. We might also add some of the other interesting stuff at [1]. — Sverdrup (talk) 21:29, 3 Mar 2004 (UTC)

The Interesting and Uninteresting Numbers section is unrelated to the rest of the article and must be moved elsewhere. It want to edit the part about 1729 but the contentious stuff is getting in the way. -- Arvindn 12:07, 5 Mar 2004 (UTC)

I have moved the paradox part to interesting number paradox. -- Arvindn 17:42, 5 Mar 2004 (UTC)

I had moved it to interesting and uninteresting numbers. But wik is reverting all attempts at changing this page. Anthony DiPierro 17:47, 5 Mar 2004 (UTC)

The one paragraph is not worth a separate article. --Wik 17:55, Mar 5, 2004 (UTC)

So put it on VfD. Also, this would be more than one paragraph if you would let me expand on it. Anthony DiPierro 17:59, 5 Mar 2004 (UTC)

Stop fighting, people. Why are you reverting Wik's edits? Adding "an interesting" isn't going to hurt anyone; for that matter, nobody would notice the difference. ugen64 22:28, Mar 5, 2004 (UTC)

"an interesting" is POV. I think it's actually rather uninteresting. Other arguments are over whether this tangentially related information should be moved to another article, and whether mention of the tongue-in-cheek proof should be included. See the discussion above. Most of it is still in dispute. Or look further at the reversions. There are many things in dispute. Anthony DiPierro 04:01, 6 Mar 2004 (UTC)

Why? Why do we have articles on simple numbers in the first place? People seem to like to put in "interesting" facts about numbers -- but ofcourse, we'll have to have a limit on it. I think that we shouldn't have articles about numbers of this magnitude in general -- but applying my personal view on the dreaded fame and importance, I think that 1729 deserves an own article. But what, Anthony, shall justify this article if not the H-R number? If we don't want that in this article, we should delete it at once. My opinion: we'll have no three or two articles on this subject, only one. Where we put it doesn't matter, but kindly don't fight so childishly over it. — Sverdrup (talk) 14:32, 8 Mar 2004 (UTC)

I'm soliciting proposals for the splitting of this page. Poll will open in one week, if no consensus can be reached and the question is ripe for consideration. During the one week of consideration these proposals may change, sometimes drastically, so be sure any responses are able to stand on their own. Anthony DiPierro 14:25, 8 Mar 2004 (UTC)

Proposals

• A proposal is made that this page be split into three. The anecdote about Interesting and uninteresting numbers will be moved there. The anecdote about the Hardy-Ramanujan number will be moved there, with a brief parenthetical reference made to it. The rest stays. Proposal is for a page similar to [2].

• I propose that we let the Wikipedians of Wikipedia:WikiProject Numbers worry about this page, and copy whatever content is suitable for other articles to the other articles, making sure to link them to/from this page. Robert Happelberg 16:45, 8 Mar 2004 (UTC)
  • I second this proposal. PrimeFan 21:48, 16 Mar 2004 (UTC)

• Proposal is made not to split the page at all. Page would look similar to [3].

• A proposal to split out the Interesting and uninteresting numbers section but leave the anecdote in the main page.

• Proposal is made to delete 1729 (number) and make interesting and uninteresting numbers the main entry for both the Ramanujan anecdote and the interesting number theorem. Then make Hardy-Ramanujan number a redirect to that, and make the link 1729 (the year) point to it. The trivia about Hair can be dropped.
  • My opinion is that the number 1729 should have its own article, and that the concept of Hardy-Ramanujan number should have its own article. I would contribute to a page on interesting and uninteresting numbers, but wouldn't be too concerned if there was no such page. PrimeFan 21:48, 16 Mar 2004 (UTC)

Discussion

• I oppose the splitting. All of the information seems on topic for this article. Jamesday 06:03, 10 Mar 2004 (UTC)

• The reason I would disagree with that is that 1729 (anecdote) is a celebrated part of the popular history of mathematics. It can stand alone quite well. Charles Matthews 10:16, 10 Mar 2004 (UTC)

• I think we should leave things the way they are, in that they work well as is. I also agree with Charles' sentiments as well. Dysprosia 10:26, 10 Mar 2004 (UTC)

• Apparently the second proposal is to not have a poll in the first place? Anthony DiPierro 13:38, 10 Mar 2004 (UTC)

• I don't see what's won by having it on a separate page rather than here. That it can stand alone is no reason to argue that it would better stand alone. The page is not getting too long yet, and as long as it does not, one should have a good reason to put something out, rather than to keep it in. Andre Engels 16:16, 11 Mar 2004 (UTC)
  • The part on interesting and uninteresting numbers has almost nothing to do with the number 1729. Anthony DiPierro 17:58, 11 Mar 2004 (UTC)

• I'd rather see it as _one_ page, maybe (for clarity) divided by one or two HRs, than to have to look around for information on basically the same topic i.e. this number (whether facts or anecdotes). --Palapala 17:54, 2004 Mar 11 (UTC)
  • I agree with you on the anecdote, but the part about interesting and uninteresting numbers really doesn't have anything to do with the number 1729. Anthony DiPierro 17:58, 11 Mar 2004 (UTC)

Why is 1729 (anecdote) separate to this page? I can't see any argument in favour of it. As Andre Engels says, the mere fact that it can stand alone doesn't mean it ought to be a separate article. --Camembert

It obviously shouldnt be. Solution: Rename the expression the HR number, consolidating the two (three articles). -SV(talk) 20:07, 12 Mar 2004 (UTC)

Charles Matthews took it upon himself to do this, and then he protected the page afterwards. Anthony DiPierro 20:18, 12 Mar 2004 (UTC)

• I suggest the fifth proposal (a single page under interesting and uninteresting numbers). The anecdote is a natural lead-in to and motivation for discussion of the paradox ("hmm, I wonder, then, if all numbers are interesting... maybe this can be proved..."); the anecdote alone isn't deserving of its own page; calling the page "1729 (number)" is awfully obscure. Jorend 16:45, 16 Mar 2004 (UTC)
  • While I agree with your suggestion, unfortunately, a recent push to delete number pages failed, and I think this proposal would suffer from the same backlash. Anthony DiPierro 17:11, 16 Mar 2004 (UTC)
    • If the topic of interesting numbers deserves an entry, it should be called "interesting numbers" or something along those lines. If people also want a 1729 (number) page, that's fine. It should be like all the other number pages (see 64 (number), a list of links). -- Jorend 19:58, 16 Mar 2004 (UTC)
      • Don't forget about number pages that draw correlations between the properties of a number, and offer analysis, such as the page on 1 and 496. Pages like these show the potential pages like the page on the number 64 have to transcend being a mere list of links. PrimeFan 21:38, 16 Mar 2004 (UTC)
        • Yes, you're right about that. The page on 1 in particular is a really good page. But, all articles should have a title that closely reflects what they're talking about. The page about 1 is about the number 1. The page I'm proposing is about the Hardy-Ramanujan anecdote and the interesting-number "proof"/"paradox". It's not about the number 1729, any more than the story of PT-109 is about the number 109. It's appropriate for the latter to link to the former; similarly here. Jorend 16:08, 18 Mar 2004 (UTC)

es:Mil setecientos veintinueve Numerao 22:06, 19 Mar 2004 (UTC)

it:Millesettecentoventinove

Thanks for making an effort to let us know about it. PrimeFan 02:01, 24 Mar 2004 (UTC)

Your welcome. Numerao 22:43, 25 Mar 2004 (UTC)

A Vote is on. -- Arvindn 15:22, 20 Mar 2004 (UTC)

On a hopefully less divisive note, is there a known date, or at least year, for the Hardy-Ramanujan anecdote? -FZ

Of course, equating "smallest" with "most negative", as opposed to "closest to zero" gives rise to solutions like -189, -1729, and further negative numbers. This unclearness demonstrates the need to define mathematical terms precisely!

I don't know if its just the exclamation mark, but that last sentence sounded somewhat pov to me, so I edited it. 141.217.41.211 19:19, 16 Jul 2004 (UTC)

I've removed this paragraph completely. It's very strange to equate "smallest" with "most negative". Unless someone has a reference for this paragraph, it's original research. Pburka (talk) 13:05, 1 July 2014 (UTC)Reply[reply]

The references to 'Futurama' in this article require clarification for the general reader.

I agree; I don't understand the "SON" thing. Benandorsqueaks 05:55, 13 December 2005 (UTC)Reply[reply]

I think that's because in the story of that show, all robots are made by Mom Corp. But I don't watch that show regularly, so I could be wrong. What time does it come on? PrimeFan 18:31, 13 December 2005 (UTC)Reply[reply]

I think it's in syndication somewhere. So far as the SON thing, I can't remember exactly which episode that is, but it fits with how Mom addresses her robots. — Laura Scudder ☎ 19:11, 13 December 2005 (UTC)Reply[reply]

Hope this response isn't too long after yours for you to see it, but I thought I'd try. The card is actually for Son #1729, i.e., Bender is the 1729^th unit that the particular manufacturing robot (his "mommy") has made. I believe it was David X. Cohen that incorrectly stated that 1729 is Bender's serial number, but it's not, so this is the best explanation I can find. Also, Mom never refers to her robots as "Son", and even if she did, it wouldn't make sense, since the card is from a robot. Additionally, in "The Farnsworth Parabox", one of the universes visited is Universe 1729. Hope I've helped. Buddy13 01:25, 31 March 2006 (UTC)Reply[reply]

While recently trawling "Articles with unsourced statements" I came across this page and was curious so had a look. Firstly I am not a mathamatician, however when I saw this I was struck by the absurdity of it. It it seems self explanatory and therefore does not need citation.

"It has occasionally been suggested that Hardy's story is apocryphal, on the grounds that he almost certainly would have been familiar with some of these features of the number.[citation needed]"

I would like to hear other thoughts on the matter--Matt 06:14, 14 January 2007 (UTC)Reply[reply]

• Oops I didnt check the external links it appears it is mentioned there The Dullness of 1729 which I believe is a valid citation --Matt 06:43, 14 January 2007 (UTC)Reply[reply]

The number 1729 can also be expressed as the sum of the cubes of the four positive numbers 10, 8, 6 and 1. Besides this feature, the number 1729 can be expressed as the sum of the squares of the following three numbers sets : [39, 12, & 8]; [37, 18, & 6]; [33, 24, & 8]; and [30, 27, & 10]. The number 1729 can also be expressed as the sum of the squares of the following four numbers sets : [35, 20, 10, & 2]; [32, 20, 17, & 4]; [32, 20, 16, & 7]; [30, 27, 8, & 6]; [28, 24, 15, & 12]; [27, 24, 18, & 10] and [26, 22, 20, & 13]. --68.193.2.168 (talk) 18:04, 29 August 2012 (UTC)Reply[reply]

Among the prime factors of 1729: 7 is the first centred hexagon, 13 is the first centred hexagram and 19 is the second centred hexagon.--DStanB (talk) 09:30, 14 October 2017 (UTC)Reply[reply]

—Sladen (talk) 10:47, 15 October 2013 (UTC)Reply[reply]

Great llamas of the Bahamas! Lugnuts ^{Dick Laurent is dead} 12:29, 15 October 2013 (UTC)Reply[reply]

I’d like to propose a minor change to the article entry under ‘Other properties’ that begins: “1729 has another mildly interesting property:”, and amend the continuation to: “…the 1729th decimal place of the transcendental number e.^[1] is the beginning of the first consecutive occurrence of all ten digits.” If this were just a minor grammatical improvement I would just go ahead and do it. However, noting that the entry already refers to ‘decimal place’, I’d like to know if editors agree with me that, despite omitting the expression ‘decimal representation’, there is still no need to add the word ‘decimal’ before the word ‘digits’.--DStanB (talk) 08:57, 14 October 2017 (UTC)Reply[reply]

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Fujiwara's theorem might be true for 10 as the base only.

1729, the Hardy-Ramanujan Number, is the smallest number which can be expressed as the sum of two different cubes in two different ways. 1729 is the sum of the cubes of 10 and 9 - a cube of 10 is 1000 and a cube of 9 is 729; adding the two numbers results in 1729. 2409:4056:18D:BCDE:2473:258A:C363:762D (talk) 04:55, 22 December 2021 (UTC)Reply[reply]

Yes, 1729 (number) already says so. I have reverted your repetition [4] of this. Things are often stated both in the lead and later but there is no reason to say it twice in the lead. PrimeHunter (talk) 07:26, 22 December 2021 (UTC)Reply[reply]

Hi I m aa Gaurav no I'm still on campus for the weekend. 2402:3A80:9CE:9455:C98C:A215:C88:4C4D (talk) 08:30, 23 December 2021 (UTC)Reply[reply]

This page describes 1729 as the "first" taxicab number. But the dedicated page on taxicab numbers says that 1729 is the second taxicab number. Because Ta(1)=2. So you might describe 1729 as the first "non-trivial" taxicab number, but I don't know if that's what a mathematician would say. Or maybe it was the first of the series to be described in the literature or something. But there's undoubtedly a factual disagreement between these pages, and I'd guess this one is the one which should be updated. Leopd (talk) 00:33, 14 November 2023 (UTC)Reply[reply]

Your suggested wording - calling it the first "nontrivial" taxicab number - is used by one of the sources this article already cites, https://mathworld.wolfram.com/Hardy-RamanujanNumber.html. So I think it's fine wording. I agree that calling it the first taxicab number seems to be incorrect per basically everyone's definitions. I'm going to make precisely the edit that you propose, right now. ExplodingCabbage (talk) 13:30, 14 November 2023 (UTC)Reply[reply]