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= February 14 =

== Lowest Fibonacci power of 10? ==

Not counting 1. What is the lowest power of 10 that is a Fibonacci number?[[User:Naraht|Naraht]] ([[User talk:Naraht|talk]]) 16:15, 14 February 2020 (UTC)
:There aren't any. [[Fibonacci number#Fibonacci primes]] says:
::The only nontrivial [[square number|square]] Fibonacci number is 144.<ref>{{Citation|title=Square Fibonacci Numbers etc | first = JHE | last = Cohn |journal= Fibonacci Quarterly |volume= 2 |year= 1964 |pages=109–13 |url= http://math.la.asu.edu/~checkman/SquareFibonacci.html}}</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}}</ref>.
:[[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 16:22, 14 February 2020 (UTC)
{{reflist-talk}}
::If I'm right, any Fibonacci number that is divisible by 10 is divisible by 610. [[User:Georgia guy|Georgia guy]] ([[User talk:Georgia guy|talk]]) 20:31, 14 February 2020 (UTC)
:::You are right. Using the indexing scheme where {{nowrap|1=''F''<sub>0</sub> = 0}}, {{nowrap|1=''F''<sub>1</sub> = 1}}, as in our article on [[Fibonacci numbers]], the following divisibility property holds:
::::If ''m'' ≥ 3, &nbsp;''m''｜''n'' &nbsp;⇔&nbsp; ''F''<sub>''m''</sub>｜''F''<sub>''n''</sub>&nbsp;.
:::So {{nowrap|&nbsp;10｜''F''<sub>''n''</sub>}} {{nowrap|&nbsp;⇔&nbsp; (2｜''F''<sub>''n''</sub>)  ∧ (5｜''F''<sub>''n''</sub>)}} {{nowrap|&nbsp;⇔&nbsp; (''F''<sub>3</sub>｜''F''<sub>''n''</sub>)  ∧ (''F''<sub>5</sub>｜''F''<sub>''n''</sub>)}} {{nowrap|&nbsp;⇔&nbsp; (3｜''n'')  ∧ (5｜''n'')}} {{nowrap|&nbsp;⇔&nbsp; 15｜''n''}} {{nowrap|&nbsp;⇔&nbsp; 610｜''F''<sub>''n''</sub>}}&nbsp;. &nbsp;--[[User talk:Lambiam|Lambiam]] 06:43, 15 February 2020 (UTC)