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%I #24 Oct 10 2019 04:09:18
%S 1,1,1,1,1,1,2,1,1,1,1,3,1,1,1,1,1,1,1,1,2,1,2,1,3,1,3,1,1,1,3,1,3,1,
%T 1,1,1,1,2,1,1,1,9,5,1,1,2,9,1,1,1,1,3,1,1,1,5,1,1,7,1,1,1,3,1,3,2,3,
%U 1,1,1,1,1,1,1,2,1,1,5,1,1,1,1,1,1,10,1,1,1,1,1,1,1,2,20,1,6,1,9,3,1,1,1,1,1,1
%N Pisano quotients: a(n) = (p-1)/k(p) if p == +- 1 mod 5, = (2*p+2)/k(p) if p == +- 2 mod 5, where p = prime(n) and k(p) = Pisano period(p).
%C Wall (1960) in Theorems 6 and 7 proved that a(n) is an integer for n >= 4. Jarden (1946) proved that the sequence is unbounded. See Elsenhans and Jahnel (2010), pp. 1-2.
%H A. Elsenhans and J. Jahnel, <a href="http://arxiv.org/abs/1006.0824">The Fibonacci sequence modulo p^2 -- An investigation by computer for p < 10^14</a>, arXiv 1006.0824 [math.NT], 2010.
%H D. Jarden, <a href="http://www.jstor.org/stable/2306239">Two theorems on Fibonacci's sequence</a>, Amer. Math. Monthly, 53 (1946), 425-427.
%H D. D. Wall, <a href="http://www.jstor.org/stable/2309169">Fibonacci series modulo m</a>, Amer. Math. Monthly, 67 (1960), 525-532.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wall-Sun-Sun_prime">Wall-Sun-Sun prime</a>
%F a(n) = (3 - L(p))/2 * (p - L(p)) / k(p), where p = prime(n), L(p) = Legendre(p|5), and k(p) = Pisano period(p) = A001175(p).
%F a(n) > 1 if and only if prime(n) is in A222413.
%t With[{p = Prime[n]}, T = Table[a = {1, 0}; a0 = a; k = 0; While[k++; s = Mod[Plus @@ a, p]; a = RotateLeft[a]; a[[2]] = s; a != a0]; k, {n, 1, 130}]; Table[L = KroneckerSymbol[p, 5]; (3 - L)/2 (p - L)/T[[n]], {n, 4, 130}]] (* after _T. D. Noe_ *)
%Y Cf. A001175, A092330, A060305, A222361, A222413.
%K nonn
%O 4,7
%A _Jonathan Sondow_, Dec 09 2017
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