Editing Carmichael's theorem (section)

{{Short description|On prime divisors in Fibonacci and Lucas sequences}}
{{about|Carmichael's theorem on [[Fibonacci number]]s and [[Lucas sequence]]s|the recursive definition of the Carmichael function|Carmichael function}}

In [[number theory]], '''Carmichael's theorem''', named after the American mathematician [[Robert Daniel Carmichael|R.&nbsp;D. Carmichael]],
states that, for any nondegenerate [[Lucas sequence]] of the first kind ''U<sub>n</sub>''(''P'',&nbsp;''Q'') with [[relatively prime]] parameters ''P'',&nbsp;''Q'' and positive discriminant, an element ''U<sub>n</sub>'' with ''n''&nbsp;≠&nbsp;1,&nbsp;2,&nbsp;6 has at least one [[prime number|prime]] divisor that does not divide any earlier one except the 12th [[Fibonacci number]] F(12)&nbsp;=&nbsp;''U''<sub>12</sub>(1,&nbsp;−1)&nbsp;=&nbsp;144 and its equivalent ''U''<sub>12</sub>(−1,&nbsp;−1)&nbsp;=&nbsp;−144.

In particular, for ''n'' greater than 12, the ''n''th [[Fibonacci number]] F(''n'') has at least one prime divisor that does not divide any earlier Fibonacci number.

Carmichael (1913, Theorem 21) [[mathematical proof|proved]] this [[theorem]].  Recently, Yabuta (2001)<ref>{{cite journal |last1=Yabuta |first1=Minoru |title=A simple proof of Carmichael's theorem on primitive divisors |journal=Fibonacci Quarterly |date=2001 |volume=39 |issue=5 |pages=439–443 |doi=10.1080/00150517.2001.12428701 |url=http://www.fq.math.ca/Scanned/39-5/yabuta.pdf |accessdate=4 October 2018}}</ref> gave a simple proof.  Bilu, Hanrot, Voutier and Mignotte (2001)<ref>{{cite journal | first1=Yuri | last1=Bilu | first2=Guillaume | last2=Hanrot | first3=Paul M. | last3=Voutier | first4=Maurice | last4=Mignotte | title=Existence of primitive divisors of Lucas and Lehmer numbers | journal=[[J. Reine Angew. Math.]] | year=2001 | volume=2001 | issue=539 | pages=75–122 | mr=1863855 | doi=10.1515/crll.2001.080 | s2cid=122969549 | url=https://hal.inria.fr/inria-00072867/file/RR-3792.pdf }}  This paper describes sequences in terms of ''P'' and ''D'' (which it calls ''a'' and ''b''); ''Q'' = (''P''<sup>2</sup> − D)/4, so when the paper talks about the sequence with (''a'',&nbsp;''b'') = (1,&nbsp;−7), that means ''P'' = 1, ''Q'' = 2.  The full list of Lucas numbers without a primitive prime divisor is ''n'' = 1, the 23 special cases listed in Table 1, and the general cases listed in Table 3.  (Tables 2 and 4 apply to the related [[Lehmer sequence]].)</ref> extended it to the case of negative discriminants (where it is true for all ''n'' > 30).