Teoremo de Carmichael

Ĉi tiu artikolo temas pri teoremo de Carmichael pri fibonaĉi-nombroj. La termino teoremo de Carmichael povas ankaŭ temi pri la rikura difino de la funkcio de Carmichael.
Estas ankaŭ nombroj de Carmichael.

En matematiko, teoremo de Carmichael, nomita pro usona matematikisto Robert Daniel Carmichael, statas ke por ĉiu n pli granda ol 12, la n-a fibonaĉi-nombro F(n) havas minimume unu priman faktoron kiu ne estas faktoro de iu ajn pli frua fibonaĉi-nombro. La nuraj esceptoj por pli malgrandaj n estas:

F(1)=1 kaj F(2)=1 kiuj ne havas primajn faktorojn
F(6)=8 kies sola prima faktoro estas 2 (kiu estas F(3))
F(12)=144 kies nuraj primaj faktoroj estas 2 (kiu estas F(3)) kaj 3 (kiu estas F(4))

Se primo p estas faktoro de F(n) kaj ne estas faktoro de iu ajn F(m) kun m<n do p estas nomata kiel karakteriza faktoro de F(n). La teoremo de Carmichael statas ke ĉiu fibonaĉi-nombro, krom la esceptoj listigitaj pli supre, havas almenaŭ unu karakterizan faktoron.

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