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Nicolò Paganini (not the violinist) was a 16 yo Italian schoolboy when he discovered that 1184 and 1210 form a pair of amicable numbers. It is in fact, the 2nd smallest such pair, and it did escape the notice of L. Euler and everyone else. Martin Gardner wrote "Although the boy probably found it by trial and error, the discovery put his name permanently into the history of number theory."

An insistent web search finds only variations of the above facts. Given the circumstances, it is likely that the original news was this incomplete, and that we may never know any more details. Hopefully I'm proven wrong, but I only risk asking for a reference to the earliest (perhaps the first?) report on this story.

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    $\begingroup$ I was about to say "how do we know it wasn't the violinist?" but the dates don't match up. $\endgroup$ Commented May 10 at 14:05
  • $\begingroup$ In the comments to my answer here , @releseabe claims to be related to Paganini. $\endgroup$
    – Spencer
    Commented May 19 at 23:44

1 Answer 1

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The original report appears in Atti della R. Accademia delle scienze di Torino, v. 2 (1866-1867) p. 362, and is signed by the Academy's secretary A. Sobrero. It is accessible through Hathi Trust and reproduced below. However, it gives no indication of Paganini's age. Dickson's History of the Theory of Numbers (1919) adds "at age 16" to Paganini's name, with reference to this note, but does not say where this detail came from. Later authors, who bother to give a reference, typically reference Dickson.

A translation is (with thanks to @nbbo2):

To make up for an oversight that occurred in the compilation of Atti Academici that gave an account of the work presented at the meeting of December 2, 1866, the following note is inserted here.

Session of December 2, 1866

The member Genocchi orally reported that he has examined the letters of Mr. B. Nicolò I. Paganini of Genoa concerning the discovery he made of two numbers known as amicable. The two numbers are 1184 and 1240 and they are, indeed, as Mr. Paganini asserted; therefore, the members thank the esteemed aforementioned Mr. Paganini for his communication, and let the two numbers he found be published in Atti Academici.

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    $\begingroup$ Italian speaker here. The translation is good. (1) apparently Paganini sent "letters" (more than one, hence the word varie) (2) sullodato is a polite way to refer to someone already mentioned, probably "The aforementioned Paganini" or "The esteemed aforementioned Paganini" if you want to capture the polite tone, would be better. $\endgroup$
    – nbbo2
    Commented May 10 at 14:02
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    $\begingroup$ Curiously, in addition to the reference to the relevant publication in Atti della R. Accad. Sc. Torino Dickson also provides this note regarding Paganini: "Cf. Cremona's Ital. transl. of Baltzer's Mathematik, pt. III". I cannot, however, find anything relevant in this book $\endgroup$
    – njuffa
    Commented May 10 at 19:43
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    $\begingroup$ I found the correct reference to Cremona in Otto Gmelin's Ph.D thesis (Halle 1917) Über vollkommene und befreundete Zahlen, p. 59. He points to part 2 of Cremona's Elementi di matematica (Genoa 1877), and sure enough, on p. 49 of that volume one finds "Il sig. PAGANINI di Genova trovò recentemente la coppia 1184, 1210." Nothing about Paganini's age there either. $\endgroup$
    – njuffa
    Commented May 11 at 8:52
  • $\begingroup$ @njuffa Dickson was personally interested in amicable numbers and published papers about them. My best guess is that he could learn this little detail from his Italian contacts by word of mouth. I can't think of how to verify it other than digging into 19th century Italian birth records. And even Dickson does not call him a "schoolboy". Could be an inference from his age and interest in numbers inserted for pedagogical inspiration by later authors. $\endgroup$
    – Conifold
    Commented May 11 at 9:07
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    $\begingroup$ @njuffa There is a mention in his 1921 paper Perfect and Amicable Numbers, but it is after even more succinct than in the History:"The second pair was discovered in 1866 by N. Paganini at the age of 16. It was missed by Euler..." Right before, he says that "a few years ago" he personally verified all amicable pairs with the smaller number up to 6232. Maybe he inquired about the missing discoverer when preparing the History for publication. $\endgroup$
    – Conifold
    Commented May 11 at 9:43

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