OFFSET
1,2
COMMENTS
For a generalized Lucas sequence {U(n)} = {(a^n - b^n)/(a - b)}, where a + b and ab are nonzero coprime integers and (a/b) is not a root of unity, a prime factor of the n-th term of some Lucas sequence U(n) is called primitive if it does not divide U(r) for any r < n.
Let P = a + b > 0, Q = ab and D = P^2 - 4Q = (a - b)^2. In the case a, b are real (equivalently, D > 0), Carmichael shows that, if n <> 1, 2, 6, then U(n) has at least one primitive prime factor not dividing D except U(3) ((P, Q, D) = (1, -2, 9), (1, -1, 5)), U(5) ((P, Q, D) = (1, -1, 5)) and U(12) ((P, Q, D) = (1, -1, 5)).
Voutier determines the cases U(n) has no primitive prime factor for n = 5, 7 <= n <= 30. For n = 5, 7 <= n <= 30, any prime factor of U(n) divides D or U_r for some r < n in the following cases:
U(5): (P, Q, D) = (1, -1, 5)*, (1, 2, -7), (1, 3, -11), (1, 4, -15)*, (2, 11, -40)*, (12, 55, -76), (12, 377, -1364),
U(7): (P, Q, D) = (1, 2, -7)*, (1, 5, -19),
U(8): (P, Q, D) = (1, 2, -7), (2, 7, -24),
U(10): (P, Q, D) = (2, 3, -8), (5, 7, -3), (5, 18, -47),
U(12): (P, Q, D) = (1, -1, 5), (1, 2, -7), (1, 3, -11), (1, 4, -15), (1, 5, -19), (2, 15, -56),
U(13), U(18), U(30): (P, Q, D) = (1, 2, -7).
(The symbol * indicates that any prime factor of the corresponding U(n) divides D)
Bilu, Hanrot and Voutier shows that these are all for n = 5, n >= 7. Hence this sequence is complete.
From Jianing Song, Feb 23 2019: (Start)
Let {U(n)} be a generalized Lucas sequence. If p is a prime U(p) has no primitive prime factor, then U(p) = +-1. In contrast, if k is a composite number such that U(k) has no primitive prime factor, then U(k) cannot be +-1. As a result, the possible solutions to U(k) = +-1 for some Lucas sequence are k = 1, 2, 3, 5, 7, 13.
From U(1) = 1, U(2) = P, U(3) = P^2 - Q, U(4) = P*(P^2 - 2*Q), U(6) = P*(P^2 - Q)*(P^2 - 3*Q) we can see that for k = 1, 2, 3, 4, 6, there are infinitely many Lucas sequences such that U(k) has no primitive prime factor. Also, for p = 2, 3, there are infinitely many Lucas sequences such that any prime factor of U(p) divides D.
This sequence lists also numbers k such that E(p) = k has no solution over the primes for some Lucas sequence {U(n)}, where E(p) is the entry point of {U(n)} modulo p, that is, the smallest m > 0 such that p divides U(m). (End)
LINKS
Y. Bilu, G. Hanrot, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers J. reine Angew. Math. 539 (2001), 75--122, a preprint version available from here.
R. D. Carmichael, On the numerical factors of the arithmetic forms a^n +- b^n, Ann. of Math., 15 (1913), 30--70.
P. M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comp. 64 (1995), 869--888.
M. Yabuta, A simple proof of Carmichael's theorem on primitive divisors, Fibonacci Quart., 39 (2001), 439--443.
EXAMPLE
If (P, Q, D) = (1, -1, 5) (giving the Fibonacci sequence), U(12) = 144 = 2^4 * 3^2, while U(4) = 3 and U(6) = 8 = 2^3. Hence U(12) with (P, Q, D) = (1, -1, 5) has no primitive prime factor and 12 belongs to this sequence.
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Tomohiro Yamada, Apr 17 2017
STATUS
approved