A336993 - OEIS

%I #12 Jan 05 2025 19:51:41

%S 1,2,3,5,8,12,13,18,30

%N List of m such that b(m) has no primitive factor, where {b(m)} is the generalized Lucas sequence defined by b(0) = 0, b(1) = 1 and b(m) = b(m-1) - 2*b(m-2) for m >= 2 (A107920).

%C A generalized Lucas sequence {U(m)} is a sequence defined by U(0) = 0, U(1) = 1 and U(m) = P*U(m-1) - Q*U(m-2) for m >= 2, where P > 0, gcd(P, Q) = 1 and (P, Q) != (1, 0), (1, 1) or (2, 1). The discriminant is D = P^2 - 4*Q.

%C A primitive factor of U(m), m > 0 is any prime that divides U(m) but does not divide U(r) for any 0 < r < m. Let's call a prime factor of U(m) "strongly primitive" if it is primitive and does not divide D.

%C Bilu, Hanrot and Voutier shows that for (P, Q, D) != (1, 2, -7), if U(m) has no strongly primitive factor, than m = 1..8, 10 or 12 (except for a few cases, m can only be 1, 2, 3, 4 or 6. See A285314). (P, Q, D) = (1, 2, -7) is the only case where U(13), U(18) and U(30) have no primitive factor.

%C This is also the only case where there are nine terms without a primitive factor. The case (P, Q, D) = (1, 3, -11) has five (m = 1, 2, 5, 6, 12). (1, -1, 5) has four (m = 1, 2, 6, 12). (1, 5, -19) has four (m = 1, 2, 7, 12). For all other Lucas sequences there can be at most three terms without a primitive factor (e.g. m = 1, 2, 6 for (3, 2, 1)).

%C Note that b(7) = 7 has only a primitive factor (7) that is not strongly primitive.

%H 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 R. D. Carmichael, <a href="https://dx.doi.org/10.2307/1967797">On the numerical factors of the arithmetic forms a^n +- b^n</a>, Ann. of Math., 15 (1913), 30--70.

%H 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 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 We have b(1) = b(2) = 1 and b(3) = b(5) = b(13) = -1, so obviously b(m) has no primitive factor if m = 1, 2, 3, 5, 13.

%e b(8) = -3 has only one prime factor 3, but 3 divides b(4) = -3, so 8 is a term here.

%e b(12) = 45 has two prime factors 3 and 5, but 3 divides b(4) = -3 and 5 divides b(6) = 5, so 12 is here.

%e b(18) = 85 has two prime factors 5 and 17, but 5 divides b(6) = 5 and 17 divides b(9) = -17, so 18 is here.

%e b(30) = -24475 has three prime factors 5, 11 and 89, but 5 divides b(6) = 5, 11 divides b(10) = -11 and 89 divides b(15) = -89, so 30 is also here.

%e According to Bilu, Hanrot and Voutier, b(m) has at least one primitive factor for any other m (and at least one strongly primitive factor if m != 7).

%Y Cf. A285314, A107920.

%K nonn,fini,full

%O 1,2

%A _Jianing Song_, Aug 10 2020