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 , the integers such that, when is written in base , the integers obtained by all

cyclicpermutation 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 ; the cyclic permutations yield the additional integers and ; and — lo and behold — we have indeed

exhibiting the desired arithmetic progression. (One also allows a leading digit being , wich can be permuted with the others, so that, for instance, , with companions and , is also a solution.)

More impressively consider ; with — in order — the progression

with common difference equal to (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 . 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…