# Story of Grothendieck's Prime Number

There is a story about Alexander Grothendieck and the "Grothendieck Prime" 57, which goes roughly as follows (cf. this wikipedia article):

In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

This quote is taken from Allyn Jackson's article "Comme Appelé du Néant— As If Summoned from the Void: The Life of Alexandre Grothendieck". Jackson refers to the story as a "legend". One can argue that the story is quite believable given Grothendieck's way of thinking (David Mumford: "He (Grothendieck) doesn’t think concretely").

Question. What (if any) is the factual basis of the story?

For instance, when/where did it happen? (In different versions it is said to have happened during or after a Grothendieck's talk.) Did anybody hear this story from somebody present at Grothendieck's talk?

My guess is that the story is just a legend, but I could be mistaken.

The story is not made up: Grothendieck did make that silly blunder, in an exchange after a talk, after being asked to be more concrete by a member of the audience. Of course this doesn't change anything to the fact that Grothendieck was one of the most profound arithmeticians of the 20th century. And indeed 57 looks a bit prime for some psychological reason :-) . Conversely many mathematicians think I'm pulling their leg when I tell them that 4999 is prime! ... I've heard this story a long time ago. I think it is true but I can't prove it since, alas, most of the protagonists are dead. Anyway, this is just an amusing but quite meaningless anecdote: a genius made a lapsus linguae. So what? On the other hand I'm quite sure that Allyn Jackson can't disprove what he (strangely) calls a legend...

Cross-posted at Mathoverflow: https://mathoverflow.net/questions/326912/story-of-grothendiecks-prime-number

• There is a similarly inane story about Kummer not being able to work out $7 \cdot 9$, or about Gauss on the number of needles on a Christmas tree. I have no idea why people continue to disseminate such stories without giving a reference - this is a very idiotic behavior even if you call the story a legend.
– user2255
Aug 2, 2017 at 8:04
• It doesn't surprise me; you can go pretty far in mathematics just using the numbers zero and one; on the other hand there's the story of Ramanujan and Hardy and the number of his 'boring' taxi-cab number, 1729; which just goes to show the different interests and capabilities of mathematicians. Aug 3, 2017 at 15:56
• Dear @MoziburUllah: I would not be surprised either, but my question is if the story is true or not. For instance, I would not be surprised if Helen of Troy did exist, but the question would be if her story is real or merely a legend. Also, for instance, I would not be surprised at all if the entire story of Grothendieck prime was invented by V.Arnold, whose "appreciation" of Bourbaki and Bourbaki-style mathematics is well known. Aug 3, 2017 at 23:02
• Dear @moishe Cohen, having looked at a volume of Bourbaki I'd go along with V.I.Arnold 'appreciation'; the point I was trying to make (and not perhaps too clearly), is that it is not neccessarily the case that a good mathematician will be good at arithmetic; the point holds in other fields, for example, Yeats is generally taken to be one of the most significant poets in the English Language, yet he was notoriously bad at spelling, in this case it's easy to check to the truth of the matter, we need only look at his manuscripts! Aug 3, 2017 at 23:25
• Anyway, I think it's a fabulous story, and even if it isn't true, it ought to be true! Aug 3, 2017 at 23:28

While it certainly does not confirm (nor refutes) Grothendieck's episode, the fact that Hermann Weyl did commit this mistake might add some 'substance' to it

The notion of prime number is of course as old and as primitive as that of the multiplication of natural numbers. Hence it is most surprising to find the distribution of primes among all natural numbers is of such a highly irregular and almost mysterious character. While on the whole the prime numbers thin out the further one gets in the sequence of numbers, wide gaps are always followed again by clusters. An old conjecture of Goldbach's maintains that there even come along again and again pairs of primes of the smallest possible difference 2, like 57 and 59.

in Weyl, H. (1951). A half-century of mathematics. The American mathematical monthly, 58(8), 523-553.

• Dear @alkchf: While this does not answer the question, it suggests a possible source of the Grothendieck Prime story: Somebody misremembered and misattributed the quote. Apr 17, 2019 at 20:41
• Was the twin prime conjecture really stated by Goldbach? Jun 19, 2019 at 16:01
• @T.Verron No, it was de Polignac. Weyl has bestowed us with two mistakes for the price of one. Feb 9, 2020 at 15:36

Theoretical mathematicians and physicists often make calculational errors. And it seems that this is the case here. The numbers that theoretical mathematicians often use are one or two with the rest having a vanishing small probability of occuring. Hence, it's not surprising they make mistakes.

Without actually being comfortable with using prime numbers on a daily basis, I would have said 57 was prime. Like Grothendieck did. Though of course, I'm not comparimg myself to him in any way. Another example is a theoretical physicist who is interested space and geometry but who often mixes up his left and right - like I do - and have just done so in the last hour or so.

• Sadly, this does not really answer the posed question. There is a difference between "Could this event have happened?" (or "Did similar events occur?") and "Did this event actually happen? If so, when and where?" Sep 26, 2021 at 14:31
• @Moishe Kohan: I'm pointing out the reasons why this story that Grothendieck didn't know what a prime number has legs. This is analysis. And history requires analysis if it is not to be just fact piled on fact or alleged facts piled on alleged facts, or just left free-standing as an alleged fact. Sep 30, 2021 at 7:21
• @Moishe Kahan: I'm not saying 'could this event have happened' but saying, if this hapoened, then this is the likely reason why. Sep 30, 2021 at 7:22
• @Moishe Kahan: And I don't consider that to be sad at all. Sep 30, 2021 at 7:22