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Nombor Fibonacci
{{pelbagai isu|{{cleanup|reason=memerlukan penterjemahan segera kerana sudah ditinggalkan sejak tahun 2008|date=Ogos 2014}}{{Terjemah|en|fabonacci number|date=Ogos 2014}}}}
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[[Image:FibonacciBlocks.svg|thumb|180px|right|Suatu ubinan dengan segi empat yang tepinya adalah nombor Fibonaci berturut-turut pada panjangnya]]
(contracted; show full)
:<math>\sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sum_{j=1}^k {F_{j}}^2} = \frac{\sqrt{5}-1}{2}.</math>

Results such as these make it plausible that a closed formula for the plain sum of reciprocal Fibonacci numbers could be found, but none is yet known. Despite that, the [[reciprocal Fibonacci constant]]
:<math>\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots</math> 

has been proved [[irrational number|irrational]] by [[Richard André-Jeannin]].

==
Primes and divisibilityNombor perdana dan kebolehbahagian==
{{main|Fibonacci primeerdana}}
A '''Fibonacci prime''' is a Fibonacci number that is [[prime number|primeerdana''' adalah nombor Fibonacci yang [[nombor perdana|perdana ]] {{OEIS|id=A005478}}. The first few are:
: 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. They must all have a prime index, excepterdana dengan beribu-ribu digit telah ditemui, tetapi tidak diketahui sama ada ia terdapat banyak tak terhingga. Semuanya mesti mempunyai indeks utama, kecuali ''F''<sub>4</sub> = 3. There are [[Arbitrarily large|arbitrarily long]] runs of [[composite number]]s and therefore also of composite Fibonacci numbers.

With the exceptions ofdapat aturan [[besar sembarangan|yang sewenang-wenangnya panjang]] bagi [[nombor komposit]] dan dengan itu termasuk juga nombor Fibonacci komposit.

Dengan pengecualian 1, 8 dand 144 (''F''<sub>0</sub> = ''F''<sub>1</sub>, ''F''<sub>6</sub> dand ''F''<sub>12</sub>) every Fibonacci number has a primesetiap nombor Fibonacci mempunyai facktor that is not autama yang bukan facktor of any smaller Fibonacci number ([[Carmichael's theoremmana-mana nombor Fibonacci yang lebih kecil ([[teorem Carmichael]]).<ref>Ron Knott, [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibtable.html "The Fibonacci numbers"].</ref>

No Fibonacci number greater than ''F''<sub>6</sub> = 8 is one greater or one less than a primeTiada nombor Fibonacci lebih besar daripada ''F''<sub>6</sub> = 8 yang lebih besar atau kurang satu daripada nuomberor perdana.<ref>Ross Honsberger ''Mathematical Gems III'' (AMS Dolciani Mathematcal Expositions No. 9), 1985, ISBN 0-88385-318-3, p. 133.</ref>

Any three consecutive Fibonacci numbers, taken two at a time, are [[relatively prime]]: that is,
:[[greatest common divisor|gcdSebarang tiga nombor Fibonacci berturut-turut, yang diambil dua pada satu masa, adalah [[perdana relatif|secara relatifnya perdana]]: itu adalah,
:[[faktor sepunya terbesar|fstb]](''F''<sub>''n''</sub>, ''F''<sub>''n''+1</sub>) = gcd(''F''<sub>''n''</sub>, ''F''<sub>''n''+2</sub>) = 1.
More generallyLebih umum,
:gcd(''F''<sub>''n''</sub>, ''F''<sub>''m''</sub>) = ''F''<sub>gcd(''n'', ''m'').</sub><ref>[[Paulo Ribenboim]], ''My Numbers, My Friends'', Springer-Verlag 2000</ref><ref>Su, Francis E., et al. [http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml "Fibonacci GCD's, please."], &#(contracted; show full)
*[http://web.archive.org/web/20070715032716/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=630&bodyId=1002 Fibonacci Numbers] at [http://web.archive.org/web/20060212072618/http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www.tools4noobs.com/online_tools/fibonacci/ Online Fibonacci calculator]

[[Kategori:Fibonacci numbers|*]]
[[Kategori:Articles containing proofs]]

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