Catalan-Dickson Conjecture/Historical Note

Historical Note on Catalan-Dickson Conjecture
first made this conjecture in $1888$, postulating that all aliquot sequences end in $1$ or a perfect number:


 * Quelques théorèmes empiriques -- Le journal Mathesis a publié, en décembre dernier, une Question proposée par M. Oltramare. Cette question m'a fait songer au théorème empirique suivant:


 * $n$ étant un nombre entier, soit $n_1$, la somme des diviseurs de $n$, inférieurs à $n$, soit $n_2$ la somme des diviseurs de $n_1$ inférieurs a $n_1$; etc. Cela posé: les nombres $n$, $n_1$, $n_2$ tendent vers une limite $\lambda$, laquel le est $1$ ou un nombre parfait.


 * Si cette proposition (vérifée sur divers exemples) est vraie, elle doit être fort difficile à démontrer.

This can be rendered in English as:


 * Some empirical theorems -- the journal Mathesis published, last December, a question posed by Mr. Oltramare. This question made me wonder about the following empirical theorem:


 * Let $n$ be an integer, let $n_1$ be the sum of divisors of $n$ less than $n$, let $n_2$ be the sum of divisors of $n_1$ less than $n_1$, and so on. This has been suggested: the numbers $n$, $n_1$, $n_2$ tend to a limit $\lambda$ which is either $1$ or a perfect number.


 * If this proposition (verified by various examples) is true, it would be very difficult to prove.

It was refined in $1913$ by to include the possibility of such a sequence ending in a sociable chain.

independently came up with the concept of aliquot sequences in $1918$, and famously made the same conjecture:

expresses the opinion, based on heuristic and experimental evidence, that there may exist some aliquot sequences which are unbounded.

demonstrated that there exist aliquot sequences of arbitrary length which are monotonically increasing.

has created such an aliquot sequence that increases for over $5000$ terms.