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(This question was posted on math.stackexchange a week ago at and still has received no answers there, so I hope it's okay to post it here too.)

In a few books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") it is said that Dirichlet figured out a proof of the unit theorem while listening to an Easter concert in the Sistine Chapel. My question is: what is the evidence for this story?

From an internet search I found that Kummer wrote on p. 343 of volume 2 of Dirichlet's collected works that Dirichlet could work on math in all kinds of situations, and then Kummer says "Als Beispiel hierfür kann ich anführen, dass er die Lösung eines schwierigen Problems der Zahlentheorie, womit er sich längere Zeit vergeblich bemüht hatte, in der Sixtinischen Kapelle in Rom ergründet hat, während des Anhörens der Ostermusik, die in derselben aufgeführt zu werden pflegt" (translation: "As an example I can say that he found the solution to a difficult problem in number theory, which he had worked on for a considerable amount of time without success, in the Sistine Chapel in Rome while he was listening to the Easter music that tends to be played there.")

Notice Kummer does not say precisely what the "difficult problem" was. Maybe it is just an oral tradition that the problem is the unit theorem, but I would like a more definitive source.

I don't read German well, but if you do then Kummer's essay on Dirichlet can be read online. It starts on and page 343 is

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I just up-voted this question, but I want to say that I don't understand why it's appropriate on MO when this what-is-the-evidence-for-what-the-historians-say-about-the-life-of-a-great-mathe‌​matician question got closed in a hot second:… – David Feldman Sep 3 '12 at 2:39
The referred-to question refers to a mathematician's non-mathematical life, this question asks about the circumstances surrounding the proof of a significant theorem, so this question is certainly MORE appropriate. – Igor Rivin Sep 3 '12 at 2:42
@David: I first posted the question on math.stackexchange since I thought that was a more appropriate first place. Nothing definitive came of that, so after a week I thought it would be okay to try it out here. – KConrad Sep 3 '12 at 3:15
Igor, my question was relevant to the intellectual life of a mathematician; in fact what I really wanted to know is whether Eisenstein's musical creations seem informed in any way by his mathematical mind. I could have asked that, but knowing whether a single example survives seemed more basic. – David Feldman Sep 3 '12 at 3:56
Just a gentle reminder: meta discussions go to meta and not the comments, in particular if the discussion is about a question other then the one at hand. – user9072 Sep 3 '12 at 4:01

In a letter to Gauss, on January 3, 1840 (Werke II), Dirichlet stated the unit theorem in the same version as he would later publish it, along with the comment that the proof was very easy and could be given on two or three pages. Indeed the proof he published in 1846 was little more than 2 pages long. In addition, the French version of his proof was contained in a letter to Liouville, and was published already in 1840. I think this shows convincingly that the problem he solved in the Sistine Chapel in 1844 was not related to the unit theorem.

On the other hand Kummer, in his talk to the memory of Dirichlet, said that Dirichlet worked out the class number formulas for the equivalence classes of forms corresponding to fields of the p-th roots of unity while he was in Italy; in fact his articles on units that Dirichlet published before his journey to Italy are directly related to problems showing up in such an investigation. When Dirichlet heard that Kummer was also working out the class number formulas for such fields, he decided not to publish anything on this subject.

Edit. I'd like to add a few quotations from Kummer's papers concerning Dirichlet's results obtained in Italy.

  • 1844, De numeris complexis.

For most numbers $\lambda$, the investigation of all units is very difficult and requires particular principles, which so far we have not yet sufficiently examined. We may skip this question, however, since we have heard that recently Lejeune-Dirichlet in Italy, where his still dwells, has worked out fundamental theorems on these complex units that we eagerly expect him to publish soon.

  • 1846, Zur Theorie der Zahlen

p. 324: This investigation on the real and ideal complex numbers is completly identical to the classification of certain related forms of degree $\lambda-1$ in $\lambda-1$ variables, of which Dirichlet has found but not yet published the main results.

p. 325: I have so far not yet studied this area of the theory of complex numbers in depth; in particular I have not yet worked out the determination of the true number of classes since I have heard by word of mouth that Dirichlet, using principles similar to those in his famous memoirs on quadratic forms, has already found this number.

  • 1846, Zerlegung der Wurzeln der Einheit:

The complete determination of those powers of ideal numbers that become real, as well as the determination of the number of inequivalent ideal numbers, requires principles that differ essentially from those given in the present memoir. We do not discuss this important question any further since, as we have already mentioned, the publication of an article by Dirichlet is imminent in which he has completely solved this question for a closely related topic.

    1. Beweis des Fermatschen Satzes

Assume that (A) the class number of $K = {\mathbb Q}(\zeta_p)$ is not divisible by $p$, and (B) that each unit in $K$ that is congruent to a rational integer modulo $p$ is actually a $p$-th power in $K$. Then the equation $x^p + y^p = z^p$ does not have nontrivial solutions in integers.

This article was presented to the Academy of Berlin by Dirichlet, who added the following remark.

The truth of the second assumption of Kummer's astute proof may be verified for each given value of $p$ using the general theory of complex units, about which I have given a few hints in the report from March 1846 and which will be published in one of the next few volumes of Crelle's journal.

After I had proved that every unit may be represented by $\frac{\lambda-3}2$ fundamental units, which is the result analogous to the general solution of Fermat's equation $x^2 - Dy^2 = \pm 1$, it was natural to follow this analogy between quadratic and these higher forms by determining the number of the latter forms by methods similar to those with which the same question in the theory of quadratic forms was solved. This investigation, to which Kummer referred at the beginning of his note, was luckily completed three years ago with the help of a new principle that was not necessary for the determination of the number of forms of the second degree, and has led to a result that is remarkable in its form and which is as simple as may be expected for a result that encompasses forms of all degrees. The expression for the number for forms of degree $\lambda-1$ that I have found, which, as may be expected by the analogy with quadratic forms, contains the $\frac{\lambda-3}2$ fundamental units, provides us, as soon as these units are known, with a method to verify assumption (A) by a rather simple numerical calculation.

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Thanks for your answer, Franz. On which page of Dirichlet's Werke does the letter start? – KConrad Sep 6 '14 at 1:18
The letter starts on p. 621 of the first volume. – Franz Lemmermeyer Jun 12 at 14:11

Kummer's remark is from his 1860 obituary of Dirichlet, so it is an early source; a recent reference that identifies the unit theorem is page 472 of "Geschichte der Universität Unter den Linden, 1810-2010: Genese der Disziplinin : die Konstitution der Universität". I made a screen shot of the text. A reference is given to Dedekind 1871. This text is online and searchable; I did not find any mention of the story there.

Franz von Krbek describes what seems to be a different story on page 11 of his 1964 book "Über Zahlen und Überzahlen", which he attributes to Frigyes [Friedrich] Riesz, stating that Dirichlet discovered his pigeonhole principle during the Easter mass in the Sixtine Chapel: Professor Dirichlet aus Berlin 1843 gerade während der Ostermesse in der sixtinischen Kapelle in Rom die Tragweite des Schufächerprinzips in der Zahlentheorie erkannte. He also writes that he attempted to find evidence for this story in Mrs. Dirichlet's correspondence to her sister, without success.

Krbek expresses doubt about the reliability of Riesz's statement, which might just be another interpretation of Kummer's 1860 remark. That seems to be the earliest source, subject to various interpretations on what the problem might have been.

For the record, here is the full quote from Krbek, with translation:

In der Antrittsrede des ungarischen Mathematikers Friedrich Riesz in Szeged aus dem Jahr 1925 heisst es, dass der Professor Dirichlet aus Berlin 1843 während der Ostermesse in der sixtinischen Kapelle in Rom die Tragweite des Schubfächerprinzips in der Zahlentheorie erkannte. Woher Riesz das hat, weiss ich nicht. In den ausführlichen Briefen der Frau Dirichlet an die Schwester steht lediglich, dass, sobald die Kirchenmusik aussetzte, ein störendes Hustenkonzert einsetzte.

In the inaugural lecture of the Hungarian mathematician Friedrich Riesz in Szeged from the year 1925 it is written, that Professor Dirichlet from Berlin realized, during the 1843 Easter mass in the Sistine chapel in Rom, the range [of applicability] of the pigeonhole principle in number theory. I do not know from where Riesz got this. In the extensive correspondence of Mrs. Dirichlet to her sister it only says that, after the church music stopped, an annoying coughing concert started.

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The line quoted from Krbek doesn't say that Dedekind discovered the pigeonhole principle in the Sistine chapel but that he recognized its range of applicability (Tragweite). Might that range include the unit theorem? – Andreas Blass Sep 3 '12 at 1:09
@Andreas: That's Dirichlet, not Dedekind. :) – KConrad Sep 3 '12 at 1:19
But it would be quite confusing if it where to refer to the unit theorem, since I assume most people will think of Dirichlet approximation theorem when hearing Dirichlet and pigeonhole. – user9072 Sep 3 '12 at 4:13
Dirichlet proved his unit theorem using the pigeonhole principle. Minkowski came later. – Franz Lemmermeyer Sep 3 '12 at 5:03
@KConrad: Thanks for the correction; too bad comments can't be edited. – Andreas Blass Sep 3 '12 at 14:09

Just a quick remark: a couple of sources cite Minkowski's 1905 Jahrsberichte essay on Dirichlet. I don't have access to the text, so don't know if he just copies the claim from Kummer.

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OK, I looked there. It is "Peter Gustav Lejeune Dirichlet und seine Bedeutung für die heutige Mathematik.", pp. 149--163 of vol. 14 of Jahresbericht der Deutschen Mathematiker-Vereinigung. On the bottom of p. 165 Minkowski writes "Es wird erzählt, daß nach langjährigen vergeblichen Bemühungen um das schwierige Problem Dirichlet die Lösung in Rom in der Sixtinischen Kapelle während des Anhörens der Ostermusik ergründet hat. Inwieweit dieses Faktum für die von manchen behauptete Wahlverwandtschaft zwischen Mathematik und Musik spricht, wage ich nicht zu erörtern." It seems like another [contd.] – KConrad Sep 3 '12 at 3:32
second hand comment, although it does link the unit theorem with the Sistine Chapel. A direct link to the text is…. I got a translation by putting it into Google translate. Anyone who knows German want to comment? (Edit: I meant p. 156 in my previous comment, not p.165.) – KConrad Sep 3 '12 at 3:38
My translation of the German: "The story is told that after many years of attempts in vain to solve the difficult problem, Dirichlet explored/found the solution in Rome in the Sistine Chapel while listening to the Easter-music. To what extent this fact supports the affinity between mathematics and music proclaimed by some, I do not dare to discuss." Some direct commentary: this is interesting in that it is in fact rather weak "es wird erzählt" is not a strong assertion, leaving open the option is is just some sort of legend. – user9072 Sep 3 '12 at 3:56

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