Cartan’s “Sur certains cycles arithmétiques”

Searching on Numdam for the papers of É. Cartan, I noticed one from 1927 entitled “Sur certains cycles arithmétiques”. Although this was not the one I was looking for, natural curiosity immediately had the better of me, and I downloaded the article, wondering what marvels could be there: an anticipation of Heegner cycles? special subvarieties of Shimura varieties?

As usual, the truth was even more surprising than such exalted expectations. Indeed, Cartan, considering “le problème de Mathématiques élémentaires proposé au dernier concours d’agrégation”, raises and solves the following question:

Classify, for any base a\geq 2, the integers n\geq 1 such that, when n is written in base a, the integers obtained by all cyclic permutation of the digits are in arithmetic progression.

This is rather surprising, since I had no idea that É. Cartan had any interest in elementary number theory; considering that he was 58 years old in 1927, I find this quite whimsical and refreshing…

Here is an example: for base 10, take n=148; the cyclic permutations yield the additional integers n_1=481 and n_2=814; and — lo and behold — we have indeed
814-481=333=481-148,
exhibiting the desired arithmetic progression. (One also allows a leading digit being 0, wich can be permuted with the others, so that, for instance, n=037, with companions 370 and 703, is also a solution.)
More impressively consider n=142857; with — in order — the progression
142857, 285714, 428571, 571428, 714285, 857142
with common difference equal to 142857 (which is also the smallest of the 6 integers.)

Cartan finds two distinct sources of such cycles, which he calls “de première [resp. de seconde] catégorie”, and classifies them, for any base a. The original problème d’agrégation asked for cycles of lengths 3 and 6 in base 10 and Cartan finds 3 cycles of length 6 and 6 of length 3. I wonder how many students managed to solve this question…

I won’t write down the solutions here — for the moment at least –, so that those readers who are interested can try their own skill…

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

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