Fibonacci sequence: Difference between revisions

Revision as of 09:14, 11 May 2025

Revision as of 15:32, 16 May 2025

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where {{math|gcd}} is the [[greatest common divisor]] function. (This relation is different if a different indexing convention is used, such as the one that starts the sequence with {{tmath|1=F_0 = 1}} and {{tmath|1=F_1 = 1}}.)

In particular, any three consecutive Fibonacci numbers are pairwise [[Coprime integers|coprime]] because both <math>F_1=1</math> and <math>F_2 = 1</math>. That is,

: <math>\gcd(F_n, F_{n+1}) = \gcd(F_n, F_{n+2}) = \gcd(F_{n+1}, F_{n+2}) = 1</math>

for every {{mvar|n}}.

EveryNot every [[prime number]] {{mvar|p}} divides a Fibonacci number that can be determined by the value of {{mvar|p}} [[modular arithmetic|modulo]]&nbsp;5. If {{mvar|p}} is congruent to 1 or 4 modulo 5, then {{mvar|p}} divides {{math|''F''_''p''−1}}, and if {{mvar|p}} is congruent to 2 or 3 modulo 5, then, {{mvar|p}} divides {{math|''F''_''p''+1}}. The remaining case is that {{math|1=''p'' = 5}}, and in this case {{mvar|p}} divides ''F_p''.

<math display=block>\begin{cases} p =5 & \Rightarrow p \mid F_{p}, \\ p \equiv \pm1 \pmod 5 & \Rightarrow p \mid F_{p-1}, \\ p \equiv \pm2 \pmod 5 & \Rightarrow p \mid F_{p+1}.\end{cases}</math>