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A285314 Numbers k such that the k-th term of some (generalized) Lucas sequence has no primitive prime factor.

%I A285314 #25 Jan 05 2025 19:51:41
%S A285314 1,2,3,4,5,6,7,8,10,12,13,18,30
%N A285314 Numbers k such that the k-th term of some (generalized) Lucas sequence has no primitive prime factor.
%C A285314 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.
%C A285314 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)).
%C A285314 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:
%C A285314 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),
%C A285314 U(7): (P, Q, D) = (1, 2, -7)*, (1, 5, -19),
%C A285314 U(8): (P, Q, D) = (1, 2, -7), (2, 7, -24),
%C A285314 U(10): (P, Q, D) = (2, 3, -8), (5, 7, -3), (5, 18, -47),
%C A285314 U(12): (P, Q, D) = (1, -1, 5), (1, 2, -7), (1, 3, -11), (1, 4, -15), (1, 5, -19), (2, 15, -56),
%C A285314 U(13), U(18), U(30): (P, Q, D) = (1, 2, -7).
%C A285314 (The symbol * indicates that any prime factor of the corresponding U(n) divides D)
%C A285314 Bilu, Hanrot and Voutier shows that these are all for n = 5, n >= 7. Hence this sequence is complete.
%C A285314 From _Jianing Song_, Feb 23 2019: (Start)
%C A285314 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.
%C A285314 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.
%C A285314 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)
%H A285314 Y. Bilu, G. Hanrot, P. M. Voutier, <a href="https://doi.org/10.1515/crll.2001.080">Existence of primitive divisors of Lucas and Lehmer numbers</a> J. reine Angew. Math. 539 (2001), 75--122, a preprint version available from <a href="https://hal.archives-ouvertes.fr/inria-00072867/">here</a>.
%H A285314 R. D. Carmichael, <a href="https://dx.doi.org/10.2307/1967797">On the numerical factors of the arithmetic forms a^n +- b^n, Ann. of Math., 15 (1913), 30--70</a>.
%H A285314 P. M. Voutier, <a href="https://doi.org/10.1090/S0025-5718-1995-1284673-6">Primitive divisors of Lucas and Lehmer sequences</a>, Math. Comp. 64 (1995), 869--888.
%H A285314 M. Yabuta, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-5/yabuta.pdf">A simple proof of Carmichael's theorem on primitive divisors</a>, Fibonacci Quart., 39 (2001), 439--443.
%e A285314 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.
%Y A285314 Cf. A001578, A058036, A246556 (smallest primitive prime factor of Fibonacci(n), Lucas(n) and Pell(n)).
%Y A285314 Cf. A086597, A086600 (number of primitive prime factors in Fibonacci(n) and Lucas(n)).
%K A285314 nonn,fini,full
%O A285314 1,2
%A A285314 _Tomohiro Yamada_, Apr 17 2017