Editing Fibonacci sequence (section)

=== Fibonacci primes ===
{{Main|Fibonacci prime}}

A ''Fibonacci prime'' is a Fibonacci number that is [[prime number|prime]]. The first few are:<ref>{{Cite OEIS|1=A005478|2=Prime Fibonacci numbers|mode=cs2}}</ref>

: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.<ref>{{citation
 | last = Diaconis
 | first = Persi
 | author-link = Persi Diaconis
 | editor1-last = Butler
 | editor1-first = Steve
 | editor1-link = Steve Butler (mathematician)
 | editor2-last = Cooper
 | editor2-first = Joshua
 | editor3-last = Hurlbert
 | editor3-first = Glenn
 | contribution = Probabilizing Fibonacci numbers
 | contribution-url = https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf
 | isbn = 978-1-107-15398-1
 | mr = 3821829
 | pages = 1–12
 | publisher = Cambridge University Press
 | title = Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham
 | year = 2018
 | access-date = 2022-11-23
 | archive-date = 2023-11-18
 | archive-url = https://web.archive.org/web/20231118192225/https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf
 | url-status = dead
 }}</ref>

{{math|''F''<sub>''kn''</sub>}} is divisible by {{math|''F''<sub>''n''</sub>}}, so, apart from {{math|1=''F''<sub>4</sub> = 3}}, any Fibonacci prime must have a prime index. As there are [[Arbitrarily large|arbitrarily long]] runs of [[composite number]]s, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

No Fibonacci number greater than {{math|1=''F''<sub>6</sub> = 8}} is one greater or one less than a prime number.<ref>{{Citation | first = Ross | last = Honsberger | title = Mathematical Gems III | journal = AMS Dolciani Mathematical Expositions | year = 1985 | isbn = 978-0-88385-318-4 | page = 133 | issue = 9}}</ref>

The only nontrivial [[square number|square]] Fibonacci number is 144.<ref>{{citation | last = Cohn | first = J. H. E. | doi = 10.1112/jlms/s1-39.1.537 | journal = The Journal of the London Mathematical Society | mr = 163867 | pages = 537–540 | title = On square Fibonacci numbers | volume = 39 | year = 1964}}</ref>  Attila Pethő proved in 2001 that there is only a finite number of [[perfect power]] Fibonacci numbers.<ref>{{Citation | first = Attila | last = Pethő | title = Diophantine properties of linear recursive sequences II | journal = Acta Mathematica Academiae Paedagogicae Nyíregyháziensis | volume = 17 | year = 2001 | pages = 81–96}}</ref> In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.<ref>{{Citation|first1=Y|last1=Bugeaud|first2=M|last2= Mignotte|first3=S|last3=Siksek|title = Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers | journal = Ann. Math.|volume = 2 | year = 2006 | pages = 969–1018 | issue = 163 | bibcode = 2004math......3046B | arxiv = math/0403046| doi = 10.4007/annals.2006.163.969|s2cid=10266596}}</ref>

1, 3, 21, and 55 are the only [[triangular number|triangular]] Fibonacci numbers, which was [[conjecture]]d by [[Verner Emil Hoggatt Jr.|Vern Hoggatt]] and proved by Luo Ming.<ref>{{Citation|first=Ming|last=Luo|title = On triangular Fibonacci numbers | journal = Fibonacci Quart. | volume = 27 | issue = 2 | year = 1989 | pages = 98–108 |doi=10.1080/00150517.1989.12429576 | url = https://www.fq.math.ca/Scanned/27-2/ming.pdf }}</ref>

No Fibonacci number can be a [[perfect number]].<ref name="Luca2000">{{citation | first=Florian | last=Luca | title=Perfect Fibonacci and Lucas numbers | journal=Rendiconti del Circolo Matematico di Palermo | year=2000 | volume=49 | issue=2 | pages=313–18 | doi=10.1007/BF02904236 | mr=1765401 | s2cid=121789033 | issn=1973-4409 }}</ref> More generally, no Fibonacci number other than 1 can be [[multiply perfect number|multiply perfect]],<ref name="BGLLHT2011">{{citation | first1=Kevin A. | last1=Broughan | first2=Marcos J. | last2=González | first3=Ryan H. | last3=Lewis | first4=Florian | last4=Luca | first5=V. Janitzio | last5=Mejía Huguet | first6=Alain | last6=Togbé | title=There are no multiply-perfect Fibonacci numbers | journal=Integers | year=2011 | volume=11a | page=A7 | url=https://math.colgate.edu/~integers/vol11a.html | mr=2988067 }}</ref> and no ratio of two Fibonacci numbers can be perfect.<ref name="LucaMH2010">{{citation | first1=Florian | last1=Luca | first2= V. Janitzio | last2=Mejía Huguet | title=On Perfect numbers which are ratios of two Fibonacci numbers | journal=Annales Mathematicae at Informaticae | year=2010 | volume=37 | pages=107–24 | url=http://ami.ektf.hu/index.php?vol=37 | mr=2753031 | issn=1787-6117 }}</ref>