Editing Perfect number (section)

== Minor results ==
All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under [[Richard K. Guy|Richard Guy]]'s [[strong law of small numbers]]:
* The only even perfect number of the form ''n''<sup>3</sup>&nbsp;+&nbsp;1 is 28 {{harv|Makowski|1962}}.<ref>{{cite journal|first=A.|last=Makowski|title=Remark on perfect numbers|journal=[[Elem. Math.]]|volume=17|year=1962|issue=5|page=109}}</ref>
* 28 is also the only even perfect number that is a sum of two positive cubes of integers {{harv|Gallardo|2010}}.<ref>{{cite journal|first=Luis H.|last=Gallardo|title=On a remark of Makowski about perfect numbers|journal=[[Elem. Math.]]|volume=65|year=2010|issue=3 |pages=121–126|doi=10.4171/EM/149|doi-access=free}}.</ref>
* The [[multiplicative inverse|reciprocals]] of the divisors of a perfect number ''N'' must add up to 2 (to get this, take the definition of a perfect number, <math>\sigma_1(n) = 2n</math>, and divide both sides by ''n''):
** For 6, we have <math>\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{1}{1} = \frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{6}{6} = \frac{1+2+3+6}{6} = \frac{2\cdot 6}{6} = 2</math>;
** For 28, we have <math>1/28 + 1/14 + 1/7 + 1/4 + 1/2 + 1/1 = 2</math>, etc.
* The number of divisors of a perfect number (whether even or odd) must be even, because ''N'' cannot be a perfect square.<ref>{{citation|title=Computational Number Theory and Modern Cryptography|first=Song Y.|last=Yan|publisher=John Wiley & Sons|year=2012|isbn=9781118188613|at=Section 2.3, Exercise 2(6)|url=https://books.google.com/books?id=eLAV586iF-8C&pg=PA30}}.</ref>
** From these two results it follows that every perfect number is an [[Ore's harmonic number]].
* The even perfect numbers are not [[trapezoidal number]]s; that is, they cannot be represented as the difference of two positive non-consecutive [[triangular number]]s. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form <math>2^{n-1}(2^n+1)</math> formed as the product of a [[Fermat prime]] <math>2^n+1</math> with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.<ref>{{Cite journal|title=Characterising non-trapezoidal numbers|first1=Chris|last1=Jones|first2=Nick|last2=Lord|journal=The Mathematical Gazette|volume=83|issue=497|year=1999|pages=262–263|doi=10.2307/3619053|jstor=3619053|publisher=The Mathematical Association|s2cid=125545112 }}</ref>
* The number of perfect numbers less than ''n'' is less than <math>c\sqrt{n}</math>, where ''c'' > 0 is a constant.<ref name="Hornfeck (1955)">{{cite journal|last=Hornfeck|first=B|title=Zur Dichte der Menge der vollkommenen zahlen|journal=Arch. Math.|year=1955|volume=6|pages=442–443|doi=10.1007/BF01901120|issue=6|s2cid=122525522}}</ref> In fact it is <math>o(\sqrt{n})</math>, using [[little-o notation]].<ref>{{cite journal|last=Kanold|first=HJ|title=Eine Bemerkung ¨uber die Menge der vollkommenen zahlen|journal=Math. Ann.|year=1956|volume=131|pages=390–392|doi=10.1007/BF01350108|issue=4|s2cid=122353640}}</ref>
* Every even perfect number ends in 6 or 28 in base ten and, with the only exception of 6, ends in 1 in base 9.<ref>H. Novarese. ''Note sur les nombres parfaits'' Texeira J. VIII (1886), 11–16.</ref><ref>{{cite book|last=Dickson|first=L. E. | author-link = L. E. Dickson|title=History of the Theory of Numbers, Vol. I|year=1919|publisher=Carnegie Institution of Washington|location=Washington|page=25|url=https://archive.org/stream/historyoftheoryo01dick#page/25/}}</ref> Therefore, in particular the [[digital root]] of every even perfect number other than 6 is 1. 
* The only [[Square-free integer|square-free]] perfect number is 6.<ref>{{cite book|title=Number Theory: An Introduction to Pure and Applied Mathematics|volume=201|series=Chapman & Hall/CRC Pure and Applied Mathematics|first=Don|last=Redmond|publisher=CRC Press|year=1996|isbn=9780824796969|at=Problem 7.4.11, p.&nbsp;428|url=https://books.google.com/books?id=3ffXkusQEC0C&pg=PA428}}.</ref>