Carmichael's theorem

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, Carmichael's theorem, named after the American mathematician <a href="/facts/Robert_Daniel_Carmichael/tz4d4Dx6">R. D. Carmichael</a>,
states that, for any nondegenerate <a href="/facts/Lucas_sequence/kceqhe6X">Lucas sequence</a> of the first kind Un(P, Q) with <a href="/facts/Relatively_prime/bnCxCXxs">relatively prime</a> parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one <a href="/facts/Prime_number/L20FLkgn">prime</a> divisor that does not divide any earlier one except the 12th <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci number</a> F(12) = U12(1, −1) = 144 and its equivalent U12(−1, −1) = −144.
In particular, for n greater than 12, the nth <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci number</a> F(n) has at least one prime divisor that does not divide any earlier Fibonacci number.
Carmichael (1913, Theorem 21) <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> this <a href="/facts/Theorem/f2Pe1FMD">theorem</a>. Recently, Yabuta (2001) gave a simple proof. Bilu, Hanrot, Voutier and Mignotte (2001) extended it to the case of negative discriminants (where it is true for all n > 30).

Carmichael's theorem open-in-new

Carmichael's theorem