# Did H. Lebesgue claim “1 is prime” in 1899? Source?

John Derbyshire, in his text "Prime obsession: Bernhard Riemann and the greatest unsolved problem in mathematics" states that

The last mathematician of any importance who did [consider the number 1 to be a prime] seems to be Henri Lebesgue, in 1899.

What is the source of this claim about H. Lebesgue (claiming 1 is prime in 1899)? I do not want to discuss the primality of one here, just the simple question: did H. Lebesgue make such a claim. In 1899 he was beginning the series of articles that became his thesis on integration, perhaps primes were used as an example... Derbyshire told me he surely had a source, but no longer has his notes for the text. Many web pages repeat this claim, often citing Derbyshire, and in one case Smith's History (1921, v. 1), which does not appear to mention Lebesgue at all.

Perhaps just an error? V.A. Lebesgue, in his 1859 "Exercices d'analyse num\'erique," listed 1 as a prime (though in articles published both before and after this, he has 2 as the first prime).

In 1914 D.N. Lehmer published the well known List of prime numbers from 1 to 10,006,721, so surely H. Lebesgue, if he made such a claim, was not the last. So my question is simply did H. Lebesgue make such a claim in 1899?

• I would not call it "claiming" I would call it "considering". – GEdgar Mar 20 '12 at 14:33
• I think it is the wrong Lebesgue. My systematic literature search a number of years ago gave Legendre as the last mathematician of note to have called $1$ a prime. Oddly enough, I have not seen his name mentioned in this context. – André Nicolas Mar 20 '12 at 14:33
• @André It would also be of interest to know the history of the structural analog, concerning whether or not the zero ring is considered to be an integral domain (iff $1$ is a prime ideal, if one is to preserve domain $\rm R/P\iff P$ prime). – Bill Dubuque Mar 20 '12 at 14:54
• I haven't looked at the article (so perhaps you know more than I do), but it's not clear to me from Lehmer's title that he's claiming that 1 is prime. – Cam McLeman Mar 20 '12 at 16:16
• D.N. Lehmer: A prime number is defined as one that is exactly divisible by no other number than itself and unity. The number 1 itself is to be considered as a prime according to this definition and has been listed as such in the table. Some mathematicians, however, prefer to exclude unity from the list of primes, thus obtaining a slight simplification in the statement of certain theorems. The same reasons would apply to exclude the number 2, which is the only even prime, and which appears as an exception in the statement of many theorems also. The number 1 is certainly ...'' – Chris K. Caldwell Mar 20 '12 at 18:51

After further research, I suspect no source other than Derbyshire's text exists. If so, this would be just a very small slip in an excellent and well written text. I know I have made similar slips.

Here is my reasoning. First, Lebesgue published his first paper in 1898, three in 1899 and then two in 1900 (all most easily found in his collected works). None of these contain any references to the prime numbers:

1. Sur l'approximation des fonctions, Bull. Sci. Math. 22 (1898), 278--287.
2. Sur la définition de l'aire d'une surface, C. R. Math. Acad. Sci. Paris 129 (1899), 870--873.
3. Sur les fonctions de plusieurs variables, C. R. Math. Acad. Sci. Paris 128 (1899), 811--813.
4. Sur quelques surfaces non réglées applicables sur le plan, C. R. Math. Acad. Sci. Paris 128 (1899), 1502--1505.
5. Sur la définition de certaines intégrales de surface, C. R. Math. Acad. Sci. Paris 131 (1900), 867--870.
6. Sur le minimum de certaines intégrales, C. R. Math. Acad. Sci. Paris 131 (1900), 935--937.

So if Lebesgue stated that one was prime in 1899, it appears not to be in a published work of his. (This does not rule out lectures, interviews, works written about him by others...)

Second, all of the Internet references to this that we checked either cite Derbyshire, or were posted well after his work. For example, a friend of mine checked Wikipedia, and this statement about Lebesgue and unity appears in the English and Dutch entry for prime, but not the French, German, Spanish, Italian, Portuguese, Polish, Russian, Czech, or Swedish. In English it was added in 2006, after Derbyshire's 2003 text. In English only it is also found in the Wikipedia page for "Henri Lebesgue":

Is this proof Derbyshire's text is the source, absolutely not. Recall Derbyshire said that he can not recall his source, but that there was one. So if you can find a source that predates his text, I'd like to know; otherwise I think it may just have been a small transcription error while written this popular text. I have done the same myself.

As for the related question: who was the last mathematician of any importance who considered the number 1 to be a prime, my student and I settle on G. H. Hardy in our draft paper (http://arxiv.org/abs/1209.2007). Hardy's A Course of Pure Mathematics, 6th edition, 1933, presented Euclid's proof that there are infinitely many primes with a sequence of primes beginning with 1:

(This was changed in the next edition.) As discussed in our draft article, there is even a remnant of Hardy listing 1 as prime in the revised 10th edition of his text published recently.

We have a list of over 125 references pertinent to the question "is one prime" collected here: http://primes.utm.edu/notes/one.pdf

• It may be worth mentioning that your draft/pre-print article with Yeng Xiong can be found on arXiv. (Which, by the way, was a very interesting and informative read!) – Willie Wong Sep 13 '12 at 10:03

perhaps you will find this useful to read through the answers on this MO page.

From doing a lot of searches, it seems as though every citation is pointing back to the book you mention.

Is there no reference to the paper from HL where he states as much?

If not, someone would have to go through his writing and/or interviews to find the actual statement as I cannot find it.

• This MO thread was already mentioned by Gerry Myerson in a comment. – t.b. Sep 7 '12 at 0:29
• Thus this answer should really be a comment. – Sasha Sep 7 '12 at 0:42