Editing Amicable numbers (section)

== Other results ==
In every known case, the numbers of a pair are either both [[Even and odd numbers|even]] or both odd. It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known.<ref>{{Cite web|url=http://sech.me/ap/news.html#20160130|title=Amicable pairs news|access-date=2016-01-31|archive-date=2021-07-18|archive-url=https://web.archive.org/web/20210718213137/https://sech.me/ap/news.html#20160130|url-status=live}}</ref> Also, every known pair shares at least one common prime [[Divisor|factor]]. It is not known whether a pair of [[coprime]] amicable numbers exists, though if any does, the [[product (mathematics)|product]] of the two must be greater than 10<sup>65</sup>.<ref>{{cite journal
 | last = Hagis | first = Peter, Jr.
 | doi = 10.2307/2004381
 | journal = Mathematics of Computation
 | mr = 246816
 | pages = 539–543
 | title = On relatively prime odd amicable numbers
 | volume = 23
 | year = 1969| issue = 107
 | jstor = 2004381
 }}</ref><ref>{{cite journal
 | last = Hagis | first = Peter, Jr.
 | doi = 10.2307/2004629
 | journal = Mathematics of Computation
 | mr = 276167
 | pages = 963–968
 | title = Lower bounds for relatively prime amicable numbers of opposite parity
 | volume = 24
 | year = 1970| issue = 112
 | jstor = 2004629
 }}</ref> Also, a pair of co-prime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula.

In 1955 [[Paul Erdős]] showed that the density of amicable numbers, relative to the positive integers, was 0.<ref>{{cite journal|last1=Erdős|first1=Paul|title=On amicable numbers|journal=Publicationes Mathematicae Debrecen|year=2022 |volume=4|issue=1–2 |pages=108–111|doi=10.5486/PMD.1955.4.1-2.16 |s2cid=253787916 |url=https://www.renyi.hu/~p_erdos/1955-03.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.renyi.hu/~p_erdos/1955-03.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

In 1968 [[Martin Gardner]] noted that most even amicable pairs sumsdivisible by 9,<ref>{{Cite journal|last=Gardner|first=Martin|title=Mathematical Games|date=1968|url=https://www.jstor.org/stable/24926005|journal=Scientific American|volume=218|issue=3|pages=121–127|doi=10.1038/scientificamerican0368-121|jstor=24926005|bibcode=1968SciAm.218c.121G|issn=0036-8733|access-date=2020-09-07|archive-date=2022-09-25|archive-url=https://web.archive.org/web/20220925113302/https://www.jstor.org/stable/24926005|url-status=live}}</ref> and that a rule for characterizing the exceptions {{OEIS|A291550}} was obtained.<ref>{{Cite journal|last=Lee|first=Elvin|date=1969|title=On Divisibility by Nine of the Sums of Even Amicable Pairs|journal=Mathematics of Computation|volume=23|issue=107|pages=545–548|doi=10.2307/2004382|jstor=2004382|issn=0025-5718|doi-access=free}}</ref>

According to the sum of amicable pairs conjecture, as the number of the [[oeis:A360054|amicable]] numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100% {{OEIS|A291422}}. Although all amicable pairs up to 10,000 are even pairs, the proportion of odd amicable pairs increases steadily towards higher numbers, and presumably there are more of them than of the even amicable pairs ([[oeis:A360054|A360054]] in [[oeis:|OEIS]]).

[[Gaussian integer]] amicable pairs exist,<ref>Patrick Costello, Ranthony A. C. Edmonds. "Gaussian Amicable Pairs." Missouri Journal of Mathematical Sciences, 30(2) 107-116 November 2018.</ref><ref>{{Cite journal|url=https://encompass.eku.edu/etd/158/|title=Gaussian Amicable Pairs|first=Ranthony|last=Clark|date=January 1, 2013|journal=Online Theses and Dissertations}}</ref> e.g. 
s(8008+3960i)	=	4232-8280i	and s(4232-8280i)	=	8008+3960i.<ref>{{Cite web|url=https://mathworld.wolfram.com/AmicablePair.html|title=Amicable Pair|first=Eric W.|last=Weisstein|website=mathworld.wolfram.com}}</ref>