Talk:Interesting number paradox

Possible Citation sources to make article better?

Some sources cited in defence on the AfD page, just putting them here so they don't get lost, no idea as to the quality

Gardner, Martin (January 1958), "A collection of tantalizing fallacies of mathematics", Mathematical games, Scientific American, 198 (1): 92–97, JSTOR 24942039

Chaitin, G. J. (July 1977), "Algorithmic information theory", IBM Journal of Research and Development, 21 (4): 350–359, doi:10.1147/rd.214.0350

Gould, Henry W. (September 1980), "Which numbers are interesting?", The Mathematics Teacher, 73 (6): 408, JSTOR 27962064

Chaitin also calls attention to its relation to an earlier paradox of Russell on the existence of a smallest undefinable ordinal (despite the fact that all sets of ordinals have a smallest element and that "the smallest undefinable ordinal" would appear to be a definition):

Russell, Bertrand (July 1908), "Mathematical logic as based on the theory of types", American Journal of Mathematics, 30 (3): 222–262, doi:10.2307/2369948, JSTOR 2369948

All on JSTOR... obviously a paywall does not stop them being valid, just anyone on the outside from using them to improve the article.

82.22.50.11 (talk) 23:41, 13 October 2022 (UTC)

39 is really interesting!

39 is interesting since it is the smallest number > 2 such that the Mertens function return 0 (see (sequence A028442 in the OEIS), I really don’t know why David Wells thought that it is uninteresting, also see [1], all natural numbers up to 80 are really interesting, the next number 81 is 3^4 thus no reason to be uninteresting, but for 82 (why 82 is interesting? Companion Pell number? Number of 6-hexes? Squarefree semiprime? Ulam number? (Only mathematics-related properties are allowed, and only base-independent properties are allowed, thus you cannot say that 82 is interesting because it is the largest atomic number of a stable isotope, and you cannot say that 82 is interesting because its multiplicative inverse has only period length 5 in decimal)) 61.224.145.154 (talk) 06:31, 22 August 2023 (UTC)

20067

Now 20067 already appears in (sequence A379570 in the OEIS), so currently what is the smallest number not appearing in an entry of an OEIS sequence? 59.126.168.120 (talk) 07:29, 4 July 2025 (UTC)

See my topic here: Talk:Interesting number paradox#Program for finding OEIS minimum number Jakevossen5 (talk) 16:40, 8 July 2025 (UTC)

Sigh

We have an IP insisting on adding in a table of undue weight, and in reaction a few people insisting that any reference to the current state of numbers that (by design) are easily determined by looking at the online sources is "WP:SELFREF" (it's not, we can mention Wikipedia when the sources mention Wikipedia), "WP:OR" (no, finding the least number that's a redirect or redlink is not an "analysis or synthesis", it's basic set theory), and "encourage[s] lurkers to create more bad articles on boring numbers" (WTF? People create bad articles anyway, and we're supposed to welcome newbies). But I give up, I'm not going to edit war over it. Anomie⚔ 11:21, 8 July 2025 (UTC)

Program for finding OEIS minimum number

I wrote this program years ago to find the minimum number if you download the OEIS data. I'll try and update the article later today, but figured it should also be linked here in case anyone else finds it useful: https://github.com/jakevossen5/oeis-min Jakevossen5 (talk) 15:42, 8 July 2025 (UTC)

`20990` is the current number based on my program, but I wonder if that is considered OR? Maybe the bash script in that repo is simple enough we can all agree it is a routine calculation? Jakevossen5 (talk) 16:30, 8 July 2025 (UTC)

I don't think whatever number happens to be the least number not currently on OEIS is relevant to the content of the article. Your script doesn't matter for that - it would still be irrelevant even if OEIS had your script built into their website. Elestrophe (talk) 09:50, 10 July 2025 (UTC)

When the article directly says for example, the smallest natural number that does not appear in an entry of the On-Line Encyclopedia of Integer Sequences (OEIS) was originally found to be 11630 on 12 June 2009. The number fitting this definition later became 12407 from November 2009 until at least November 2011., the number currently fitting this definition seems eminently relevant. Possibly more relevant than what the value was from November 2009–2011. Anomie⚔ 11:24, 10 July 2025 (UTC)

I agree with Anomie⚔, for all of the proposed solutions to this problem we try to maintain up to date numbers, why would we not want to do the same for this one? It was relevant when it is published by the person who came up with the theory, I think it would be relevant if this was published by OEIS. Jakevossen5 (talk) 16:50, 10 July 2025 (UTC)

It is original research and it is forbidden in Wikipedia articles, whose content should be based entirely on published sources. If someone else wants to publish an article stating that, at a certain date, a certain number was the smallest not to appear, then we can use that. We should not be calculating these things ourselves. —David Eppstein (talk) 17:55, 10 July 2025 (UTC)

Application of simple mathematics falls under Wikipedia:No original research#Routine calculations. Anomie⚔ 18:19, 10 July 2025 (UTC)

"Routine calculations" refer to simple numerical calculations, like that it is one week between July 3 and July 10, that anyone reasonably competent could be expected to verify merely by looking at it. They do not refer to "combining material from multiple sources to state or imply a conclusion not explicitly stated by any of the sources", something that is explicitly forbidden by WP:NOR, and is exactly what you are pushing to do here. —David Eppstein (talk) 18:29, 10 July 2025 (UTC)

> Routine calculations do not count as original research, provided there is consensus among editors that the results of the calculations are correct, and a meaningful reflection of the sources

It doesn't state the maximum complexity of the calculations, as long as we can agree it is correct.

The 2014 article actually does not explain how they calculated it or provide any way to verify its work, probably because they assume anyone interested is competent enough to download a text file of comma separated values, loop over each number, and stop when that number isn't found in a text file. If this was something that wasn't a routine calculation, I would expect the article to describe how they calculated it. Jakevossen5 (talk) 19:57, 10 July 2025 (UTC)

It's not a calculation at all. It's synthesizing material from multiple Wikipedia or OEIS pages. It is the very definition of WP:SYN. And when applied to Wikipedia rather than OEIS, it also violates WP:CIRCULAR, the use of Wikipedia itself as a source for Wikipedia content. —David Eppstein (talk) 20:04, 10 July 2025 (UTC)

You're full of mis-cited policies today, aren't you? Take a closer read of WP:CIRCULAR, specifically where it says An exception is allowed when Wikipedia itself is being discussed in the article. These may cite an article, guideline, discussion, statistic, or other content from Wikipedia (or a sister project) to support a statement about Wikipedia. Anomie⚔ 20:08, 10 July 2025 (UTC)

That paragraph continues, "Wikipedia or the sister project is a primary source in this case and may be used following the policy for primary sources. Any such use should avoid original research...". I don't think this use would be legitimate in that regard. Wikipedia is not the place to write a scholarly study of Wikipedia. Stepwise Continuous Dysfunction (talk) 22:37, 10 July 2025 (UTC)

No one here is proposing writing a scholarly study of Wikipedia. What we do have are a few people who have decided WP:IDONTLIKEIT and are bending any policy they can find to try to support that. Anomie⚔ 10:53, 11 July 2025 (UTC)

I think doing this calculation with Wikipedia would be exactly as legitimate under CALC/OR/SYN as doing it with any other site, and the CIRCULAR invocation is thus spurious, and that while I don't want to specifically give a number to the sorites paradox as to how many steps a calculation must take before it stops being routine from length alone, checking hundreds of pages is probably pushing it and going through the whole OEIS more so. Sesquilinear (talk) 07:43, 14 July 2025 (UTC)

No, that's not at all what I'm pushing to do here. Wikipedia:No original research#Routine calculations in fact states the opposite of what you claim: Mathematical literacy may be necessary to follow a "routine" calculation, particularly for articles on mathematics or in the hard sciences. That's quite far from your anyone reasonably competent could be expected to verify merely by looking at it. Nor is this taking data from multiple sources: it's in fact taking data from exactly one source, OEIS. And not even more than one URL as you erroneously claim above, only https://oeis.org/stripped.gz. Anomie⚔ 20:06, 10 July 2025 (UTC)

When the assertion "the smallest integer not appearing in <this version> of OEIS is <that number>" does not appear in any reliably published text, it cannot be verified by routine calculation, since the verification requires (1) to have an access to <this version> of OEIS, when it is not the current version (OEIS changes almost every day) (2) to have access to the script and being able to verify its correctness, or to be able to write an equivalent script. None of these two abilities can be qualified of "routine calculation" even for a professional mathematician. So an assertion such as the one that I have quoted is definitively WP:OR, and I fully agree with David. D.Lazard (talk) 21:02, 10 July 2025 (UTC)

The OEIS data is hosted on GitHub here: https://github.com/oeis/oeisdata, so you can verify it at any point in time. And I think it is disingenuous that the idea that writing a while loop that increments a number and checks for it in a file is a very big barrier, this could be one of the very first projects in a CS 101 course, much less "advanced mathematics" that CALC says is acceptable.

I tried to search for other pages that have had debates around routine calculations, and some people have come up with the metric "if anyone can cast doubt on the method or the result" then it isn't routine. In this case, we aren't arguing about weather or not 20990 is the smallest number, or if the calculation is valid. If this wasn't a "routine calculation" there should be room for interpretation of if the calculation is correct, and I don't think that is the case here (this would be extremely easy to prove wrong by finding a smaller number than 20990) Jakevossen5 (talk) 21:59, 10 July 2025 (UTC)

I don't think that a problem that could be a homework assignment in a college course is a "routine calculation". Would that calculation be beyond the skills of many people reading this article? Yes, I'd say so. It's not like converting feet to meters, or Fahrenheit to Celsius.

For that matter, everything prior to 2023 was sourced to a blog post by a random PhD student. If that is indeed the best source for the material, the material is not actually important enough to include here. I have trimmed it to just what the Scientific American item can support. Stepwise Continuous Dysfunction (talk) 22:58, 10 July 2025 (UTC)

> I don't think that a problem that could be a homework assignment in a college course is a "routine calculation"

CALC makes it clear that advanced mathematics may be required to follow the calculation, and I am sure there are corners of Wikipedia that include math calculations I can't follow because I don't have the skills, but the community can agree that the calculation is correct. The bar shouldn't be "can anyone who comes to this page perform this calculation", otherwise there wouldn't be the line about advanced mathematics knowledge required. It seems to be more interested in if the people in the talk page can agree it is correct, and again, it seems like everyone here is agreement that 20990 is the result. Jakevossen5 (talk) 23:08, 10 July 2025 (UTC)

The line isn't "can anyone who comes to the page perform this calculation". Anybody can arrive at any page, in principle. The question is whether the calculation is radically out of step with the likely audience. The guideline says that "Mathematical literacy may be necessary", not that computations requiring the reader to download and parse a dataset are hunky-dory. And beyond that, mere correctness is not enough to justify inclusion. Is scraping the OEIS and sifting through it with a script a "meaningful reflection"? No.

Saying more than what the one halfway decent source does would be belaboring a curiosity. Stepwise Continuous Dysfunction (talk) 23:31, 10 July 2025 (UTC)

would be belaboring a curiosity. I think now we're getting to the real reason for the opposition here. You think the whole paragraph is "a curiosity" that doesn't deserve to be mentioned at all, but since there are sources for it the best you can do is ignore the plain language of WP:CALC to try to exclude finding an updated number from the primary source. Anomie⚔ 11:00, 11 July 2025 (UTC)

The "curiosity" is the paradox itself. The evolution over the time of a value provided by an evolving data base is definitely outside the scope of the article. On the opposite, providing such details can be misleading by hidden the meaning of the paragraph behind unneeded data. It is the first reason of my opposition.

About sourcing: self published sources are not allowed in Wikipedia. The script is a self-published source. So, its results have not their place in Wikipedia. Similarly, if 20,067 would not be in Bischoff's paper, it would not be sourced, and must be changed into some value.

About WP:CALC: My interpretation is that WP:CALC can be used only if an expert of the subject of the article can verify the truth of the assertion, simply by reading, or with an elementary use of a pocket calculator. This far to be the case here, at least because the script involves programming competences that are far to be relevant to the subject of the article.

About WP:Verifiability: To verify a sentence such that "At <this date>, the least integer not appearing in OEIS is <this value>, one must access to OEIS data base at <this date>. I believe that it is impossible, if <this date> is not the current date. If it is possible, this requires a very strong OEIS expertise. Also, the source of the script is not publicly available. So, readers cannot verify it. In particular, it is unclear whether 20,067 is distinguished from 20, 067, whether negative numbers are disregarded or taken as their absolute value, whether numerical parts of OEIS labels such as A1234 are considered or disregarded. This can change dramatically the output of the script. D.Lazard (talk) 12:21, 11 July 2025 (UTC)

The scripts are available from the link up at the top of the thread, so that isn't an issue. What's more significant is that checking one of them requires knowing Rust and understanding the author's oeis-utils library, and running the other requires jumping through some sudo snap install ripgrep hoops. That's something beyond "mathematical literacy". Call me old-fashioned, but "mathematical literacy" means being able to do algebra, not figuring out why you're getting an error message about a package being published using classic confinement. And then, even if one goes through all the steps to become confident in the code, the question remains: if a number hasn't been published anywhere, and can only be produced by a database-scraping computation, is it really worth including? Stepwise Continuous Dysfunction (talk) 16:43, 11 July 2025 (UTC)

self published sources are not allowed in Wikipedia That's incorrect. Never mind that the script isn't being used as a source in the first place, it's illustrating a means to perform a calculation. And never mind that the script isn't actually necessary to perform the calculation.simply by reading, or with an elementary use of a pocket calculator Wrong again, it doesn't say any such thing. I believe that it is impossible, if <this date> is not the current date. archive.org exists, if necessary. I doubt it's really necessary, there are plenty of other cases where we say "as of X date, website Y said Z" without worrying that tomorrow the website may be updated. People just update the article periodically with the updated information, and the "as of" just indicates when that was last checked. In particular, it is unclear whether 20,067 is distinguished from 20, 067, whether negative numbers are disregarded or taken as their absolute value, whether numerical parts of OEIS labels such as A1234 are considered or disregarded. Then write a different script. It's really not hard, basic data processing. You could even do it by hand if you wanted to take the time. Anomie⚔ 00:41, 12 July 2025 (UTC)

No, I don't think "the whole paragraph is "a curiosity" that doesn't deserve to be mentioned at all". Please don't put words into my mouth. I said that a blog post by a random graduate student is not a very good source. That's just standard practice. I trimmed the paragraph to what could be justified from the Scientific American story. I think what's left stands up pretty well. From everything I've seen, it doesn't need expansion, but it doesn't need further cutting either.

I don't see anything in the "plain language of WP:CALC" that justifies the exercise being gone through here. That section requires "a meaningful reflection of the sources", which isn't being done here at all. Its explicit examples are "adding numbers, converting units, or calculating a person's age", which this obviously isn't. It says "Mathematical literacy may be necessary" in some cases, but wrangling a git repository and checking the accuracy of a Rust program and a bash script while accounting for all the oddities of OEIS data formatting is not mere "mathematical literacy". Stepwise Continuous Dysfunction (talk) 16:22, 11 July 2025 (UTC)

It's not necessary to wrangle a git repository or use a Rust program. You could even do it manually, it would just take you a long time and be more error prone.

The data file is very regularly formatted: Ignore the lines at the start beginning with a # character. Each line remaining contains one sequence from OEIS, formatted as a series of comma-separated fields, with the first field being the sequence ID, the subsequent fields being the integers (represented using digits 0–9, with a preceding - if it's negative) in the start of the sequence, and finally an empty field at the end of the line.

In mathematical terms, you define $A$ as the subset of nonnegative integers that do not appear in any truncated sequence in the data file, and use the well-ordering principle to identify the least element. That seems very amenable to WP:CALC. If you don't think so, what specifically is too complex about it, besides that you personally don't know how to deal with a data file containing 16,991,277 integers to apply the set definition? Anomie⚔ 01:19, 12 July 2025 (UTC)

Coming here from NORN, and taking no position on whether the result is important enough to include or not, I don't really think the number of operations (rather than the type) that need to be performed should necessarily be considered defining for how routine a calculation is considered. Any educated person with access to a source could perform a finite, but arbitrarily large number of basic arithmetic operations (e.g., adding a million two digit numbers together) and similarly, I don't think you would need to be college educated to mechanically eliminate numbers that exist in any of an arbitrarily large number of arbitrarily long lists. The more effort it takes to arrive at the number, the less I would expect it to be appropriate to include, but there's no need for specialised knowledge to verify it at least in my opinion. Alpha3031 (t • c) 13:24, 11 July 2025 (UTC)

"It was relevant when it is published by the person who came up with the theory..." There's no "theory" here. Nobody "came up with" one. It's idle entertainment that mathematicians do when they're not actually working. Stepwise Continuous Dysfunction (talk) 23:35, 10 July 2025 (UTC)

This would obviously be original research. It's also trivia. The original value of the first-OEIS-uninteresting number is already trivia, and it's mentioned in this article in what's essentially a parenthetical aside. An ever-expanding table of such numbers would be ludicrously undue second-degree trivia. If you want to host a website of uninteresting numbers, please go ahead. And if it becomes the subject of independent significant coverage, Wikipedia can host an article about your project. pburka (talk) 23:08, 11 July 2025 (UTC)

As far as I can tell, only 59.126.168.120 wants a table of historical values for the number. All I'd like to see is "as of <recent date>, the value was now N", since it's easy to determine the value from the source and wondering what the current value is does seem like a natural question a reader might ask upon reading the paragraph. Anomie⚔ 01:24, 12 July 2025 (UTC)

It might be the kind of thing a reader could be curious about, but it's still trivia and original research. Such a statement would be forever tagged with {{citation needed}}, because there's nothing you can cite as the source. If I write that JFK was 46 years old when he was shot, I can cite sources showing his birth and death years, and trust the reader to understand that routine calculation I performed. What would you cite as your source for this factoid, and how would a reader verify it? pburka (talk) 02:18, 12 July 2025 (UTC)

I would propose something like "as of July 2025, the number was 20990" with a footnote that provides an archive.org link to the data source, and either a bash one-liner which I am sure we could come up with or something along the lines "calculated by iterating over all positive integers until you find one not present in the file" or something like that. We could use grep, it is just slower than ripgrep. And I would accept that not every reader would be able to reproduce this, just as I cannot reproduce high level math proofs in other CALCs. Jakevossen5 (talk) 03:55, 12 July 2025 (UTC)

I'd recommend the textual description. It's more likely to be understood than something like

declare -A IN_FILE=(); while IFS= read -r N; do IN_FILE[$N]=1; done < <( zcat stripped.gz | sed -e '/^#/d; s/,/\n/gp' | sed -n -E 's/^[[:space:]]+|[[:space:]]+$//g; /^[0-9]+$/p' ); for (( I=0; ${IN_FILE[$I]}; I++ )); do :; done; echo "First nonnegative integer not in file is $I"

or

use PerlIO::gzip; %inFile=(); open IN, "<:gzip", "stripped.gz" or die "Failed to open file: $!"; while(<IN>) { next if /^#/; for my $n (split /,/){ $inFile{$1}++ if $n =~ /^\s*(\d+)\s*$/; } } for($i=0; $inFile{$i} // 0; $i++){} print "First nonnegative integer not in file is $i\n";

or

fh = io.popen( "zcat stripped.gz", "r" )
inFile={}
for line in fh:lines() do
  if line:sub( 0, 1 ) ~= "#" then
    for n in line:gmatch( "[^,]+" ) do
      n = n:gsub( "^%s+", "" ):gsub( "%s+$", "" )
      if n:match( "^%d+$" ) then
        inFile[n] = true
      end
    end
  end
end
i = 0
while inFile[ tostring( i ) ] do
  i=i+1
end
print( "First nonnegative integer not in file is " .. i )

whichever method and language you'd like. Anomie⚔ 12:54, 12 July 2025 (UTC)

Clearly not original research, if you can not understand a 2 lines of script, or can not write such a very tiny script to solve a totally trivial maths problem to extract the smallest missing number from a list, then you should not be here as an editor. In this case just read and study the Wikipedia as 99.9 percent of people do this. Yes, maths is a really hard subject on professional PHD level, I can not and to tell the truth could not recommend it to everybody. Say in average there are 150 maths papers per day on arxiv. Not everybody has any maths sense, you should accept it. But no one is professional at every field, I am very weak at cooking, but very good at maths and programming. Period. 82.131.139.3 (talk) 02:14, 12 July 2025 (UTC)

There is no need to insult the technical ability of editors who disagree with you regarding Wikipedia policy. Elestrophe (talk) 02:26, 12 July 2025 (UTC)

While I find the claims of "it's original research" fallacious and either severely misguided or a disguise for "I don't like it but I need a better excuse to call for its removal", you go too far in claiming that anyone who can't write a short computer program is not qualified to be an editor. Anomie⚔ 11:44, 12 July 2025 (UTC)

I have no particular interest in whether we update this statistic, or even whether we include it in the article. But I feel very strongly that WP:CALC includes this type of calculation. There are so many math articles that rely on complicated calculations and there’s no good way to do this without doing them yourself: different sources will phrase things differently and use different bases and so forth. CRGreathouse (t | c) 14:09, 12 July 2025 (UTC)

I don't think this is the same thing: isn t a calculation at all. It's an iterative search of a database to find a minimum value. pburka (talk) 15:47, 12 July 2025 (UTC)

It's a straightforward application of the well-ordering principle. Set theory is part of mathematics. Anomie⚔ Anomie⚔ 23:56, 12 July 2025 (UTC)

Much of the above discussion has been about the original research policy, but my objection remains that including this number would be giving undue weight to the OEIS example. Remember that this is an article about the interesting number paradox in philosophy, not about the OEIS. Elestrophe (talk) 00:22, 13 July 2025 (UTC)

Agree, the article itself undercuts the importance of the OEIS example by implying that aside from an extended A000027 including every integer, extensions of the other entries may cover "the smallest natural number that does not appear in an [OEIS] entry," making the result of this script unnecessary even before considering whether it is original research. ViridianPenguin🐧 (💬) 02:37, 18 July 2025 (UTC)