Editing Aliquot sequence (section)

== Catalan–Dickson conjecture ==
An important [[conjecture]] due to [[Eugène Charles Catalan|Catalan]], sometimes called the Catalan–[[Leonard Eugene Dickson|Dickson]] conjecture, is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers.<ref>{{MathWorld | urlname=CatalansAliquotSequenceConjecture | title=Catalan's Aliquot Sequence Conjecture}}</ref> The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The first five candidate numbers are often called the '''Lehmer five''' (named after [[Derrick Henry Lehmer|D.H. Lehmer]]): [[276 (number)|276]], 552, 564, 660, and 966.<ref>{{cite web|url=http://www.aliquot.de/lehmer.htm|title=Lehmer Five|first=Wolfgang|last=Creyaufmüller|date=May 24, 2014|access-date=June 14, 2015}}</ref> However, 276 may reach a high apex in its aliquot sequence and then descend; the number 138 reaches a peak of 179931895322 before returning to 1.

[[Richard K. Guy|Guy]] and [[John Selfridge|Selfridge]] believe the Catalan–Dickson conjecture is false (so they conjecture some aliquot sequences are [[bounded function|unbounded]] above (i.e., diverge)).<ref>A. S. Mosunov, [http://www.cs.uleth.ca/~hadi/2016-09-29-aliquot_sequences.pdf What do we know about aliquot sequences?]</ref>