Editing Fibonacci sequence (section)

=== Prime divisors ===
With the exceptions of 1, 8 and 144 ({{math|1=''F''<sub>1</sub> = ''F''<sub>2</sub>}}, {{math|''F''<sub>6</sub>}} and {{math|''F''<sub>12</sub>}}) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number ([[Carmichael's theorem]]).<ref>{{Citation | first = Ron | last = Knott | url = http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibtable.html | title = The Fibonacci numbers | publisher = Surrey | place = UK}}</ref> As a result, 8 and 144 ({{math|''F''<sub>6</sub>}} and {{math|''F''<sub>12</sub>}}) are the only Fibonacci numbers that are the product of other Fibonacci numbers.<ref>{{Cite OEIS|1=A235383|2=Fibonacci numbers that are the product of other Fibonacci numbers|mode=cs2}}</ref>

The divisibility of Fibonacci numbers by a prime {{mvar|p}} is related to the [[Legendre symbol]] <math>\bigl(\tfrac{p}{5}\bigr)</math> which is evaluated as follows:
<math display=block>\left(\frac{p}{5}\right) = \begin{cases} 0 & \text{if } p = 5\\ 1 & \text{if } p \equiv \pm 1 \pmod 5\\ -1 & \text{if } p \equiv \pm 2 \pmod 5.\end{cases}</math>

If {{mvar|p}} is a prime number then
<math display=block> F_p \equiv \left(\frac{p}{5}\right) \pmod p \quad \text{and}\quad F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p.</math><ref>{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | year = 1996 | title = The New Book of Prime Number Records | place = New York | publisher = Springer | isbn = 978-0-387-94457-9 | page = 64}}</ref>{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.25–28 | pp = 73–74}}

For example,
<math display=block>\begin{align}
\bigl(\tfrac{2}{5}\bigr) &= -1, &F_3  &= 2, &F_2&=1, \\
\bigl(\tfrac{3}{5}\bigr) &= -1, &F_4  &= 3,&F_3&=2, \\
\bigl(\tfrac{5}{5}\bigr) &= 0, &F_5  &= 5, \\
\bigl(\tfrac{7}{5}\bigr) &= -1, &F_8  &= 21,&F_7&=13, \\
\bigl(\tfrac{11}{5}\bigr)& = +1, &F_{10}&  = 55, &F_{11}&=89.
\end{align}</math>

It is not known whether there exists a prime {{mvar|p}} such that

<math display=block>F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod{p^2}.</math>

Such primes (if there are any) would be called [[Wall–Sun–Sun prime]]s.

Also, if {{math|''p'' ≠ 5}} is an odd prime number then:{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.28 | pp = 73–74}}
<math display=block>5 {F_{\frac{p \pm 1}{2}}}^2 \equiv \begin{cases}
\tfrac{1}{2} \left (5\bigl(\tfrac{p}{5}\bigr)\pm 5 \right ) \pmod p & \text{if } p \equiv 1 \pmod 4\\
\tfrac{1}{2} \left (5\bigl(\tfrac{p}{5}\bigr)\mp 3 \right ) \pmod p & \text{if } p \equiv 3 \pmod 4.
\end{cases}</math>

'''Example 1.''' {{math|1=''p'' = 7}}, in this case {{math|1=''p'' ≡ 3 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{7}{5}\bigr) = -1: \qquad \tfrac{1}{2}\left(5 \bigl(\tfrac{7}{5}\bigr)+3 \right ) =-1, \quad \tfrac{1}{2} \left(5\bigl(\tfrac{7}{5}\bigr)-3 \right )=-4.</math>
<math display=block>F_3=2 \text{ and } F_4=3.</math>
<math display=block>5{F_3}^2=20\equiv -1 \pmod {7}\;\;\text{ and }\;\;5{F_4}^2=45\equiv -4 \pmod {7}</math>

'''Example 2.''' {{math|1=''p'' = 11}}, in this case {{math|1=''p'' ≡ 3 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{11}{5}\bigr) = +1: \qquad  \tfrac{1}{2}\left( 5\bigl(\tfrac{11}{5}\bigr)+3 \right)=4, \quad \tfrac{1}{2} \left(5\bigl(\tfrac{11}{5}\bigr)- 3 \right)=1.</math>
<math display=block>F_5=5 \text{ and } F_6=8.</math>
<math display=block>5{F_5}^2=125\equiv 4 \pmod {11} \;\;\text{ and }\;\;5{F_6}^2=320\equiv 1 \pmod {11}</math>

'''Example 3.''' {{math|1=''p'' = 13}}, in this case {{math|1=''p'' ≡ 1 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{13}{5}\bigr) = -1: \qquad \tfrac{1}{2}\left(5\bigl(\tfrac{13}{5}\bigr)-5 \right) =-5, \quad \tfrac{1}{2}\left(5\bigl(\tfrac{13}{5}\bigr)+ 5 \right)=0.</math>
<math display=block>F_6=8 \text{ and } F_7=13.</math>
<math display=block>5{F_6}^2=320\equiv -5 \pmod {13} \;\;\text{ and }\;\;5{F_7}^2=845\equiv 0 \pmod {13}</math>

'''Example 4.''' {{math|1=''p'' = 29}}, in this case {{math|1=''p'' ≡ 1 (mod 4)}} and we have:
<math display=block>\bigl(\tfrac{29}{5}\bigr) = +1: \qquad \tfrac{1}{2}\left(5\bigl(\tfrac{29}{5}\bigr)-5 \right)=0, \quad \tfrac{1}{2}\left(5\bigl(\tfrac{29}{5}\bigr)+5 \right)=5.</math>
<math display=block>F_{14}=377 \text{ and } F_{15}=610.</math>
<math display=block>5{F_{14}}^2=710645\equiv 0 \pmod {29} \;\;\text{ and }\;\;5{F_{15}}^2=1860500\equiv 5 \pmod {29}</math>

For odd {{mvar|n}}, all odd prime divisors of {{math|''F''<sub>''n''</sub>}} are congruent to 1 modulo 4, implying that all odd divisors of {{math|1=''F''<sub>''n''</sub>}} (as the products of odd prime divisors) are congruent to 1 modulo 4.{{Sfn | Lemmermeyer | 2000 | loc = ex. 2.27 | p = 73}}

For example,
<math display=block>F_1 = 1,\ F_3 = 2,\ F_5 = 5,\ F_7 = 13,\ F_9 = {\color{Red}34} = 2 \cdot 17,\ F_{11} = 89,\ F_{13} = 233,\ F_{15} = {\color{Red}610} = 2 \cdot 5 \cdot 61.</math>

All known factors of Fibonacci numbers {{math|''F''(''i'')}} for all {{math|''i'' < 50000}} are collected at the relevant repositories.<ref>{{Citation | url = https://mersennus.net/fibonacci/ | title = Fibonacci and Lucas factorizations | publisher = Mersennus}} collects all known factors of {{math|''F''(''i'')}} with {{math|''i'' < 10000}}.</ref><ref>{{Citation | url =http://fibonacci.redgolpe.com/ | title = Factors of Fibonacci and Lucas numbers | publisher = Red golpe}} collects all known factors of {{math|''F''(''i'')}} with {{math|10000 < ''i'' < 50000}}.</ref>