A051830 -id:A051830 - cp's OEIS frontend

A086937 Number of distinct zeros of x^2-x-1 mod prime(n).

0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2, 2

Offset: 1

N. J. A. Sloane, Sep 23 2003

nonn

For the prime modulus 5, the polynomial can be factored as (x+2)^2, showing that x=3 is a zero of multiplicity 2. The discriminant of the polynomial is 5. Also note how this sequence is related to the Fibonacci sequence A051830; for n>3, a(n) = 2*A051830(n). - T. D. Noe, Aug 13 2004

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.

Cf. A086965, A086966, A086967.

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 105}] (* T. D. Noe, Sep 24 2003 *)

If p = prime(n), a(n) = A080891(p) + 1.

Corrected and extended by T. D. Noe, Sep 24 2003

A051831 a(n) = Fibonacci(prime(n)) mod prime(n), where prime(n) is the n-th prime.

Original entry on oeis.org

1, 2, 0, 6, 1, 12, 16, 1, 22, 1, 1, 36, 1, 42, 46, 52, 1, 1, 66, 1, 72, 1, 82, 1, 96, 1, 102, 106, 1, 112, 126, 1, 136, 1, 1, 1, 156, 162, 166, 172, 1, 1, 1, 192, 196, 1, 1, 222, 226, 1, 232, 1, 1, 1, 256, 262, 1, 1, 276, 1, 282, 292, 306, 1, 312, 316, 1, 336, 346, 1, 352, 1

Offset: 1

Jud McCranie, Dec 11 1999

nonn

Terms are 1 when prime(n) == 1 or 4 mod 5, terms are prime(n)-1 when prime(n) == 2 or 3 mod 5.

In general, it appears that Fibonacci(k*p) mod p = Fibonacci(k) or p-Fibonacci(k) for prime p > Fibonacci(k). For example Fibonacci(8*29) mod 29 = 21. - Gary Detlefs, May 28 2014

			prime(3) = 5, fibonacci(5) = 5 == 0 mod 5.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
R. Peele and P. Stanica, Matrix powers of column-justified Pascal triangles and Fibonacci sequences, arXiv:math/0010186 [math.CO], 2000.

Cf. A000045, A003631, A045468, A051834, A051830, A236395.

p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, ithprime(n)$2)[1, 2]:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 10 2015

Mod[Fibonacci[Prime[#]],Prime[#]]&/@Range[75] (* Harvey P. Dale, Jan 14 2011 *)

vector(80, n, fibonacci(prime(n)) % prime(n)) \\ Michel Marcus, Jul 15 2015

A051834 Fibonacci(Pn-1) mod Pn, where Pn is the n-th prime.

Original entry on oeis.org

1, 1, 3, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1

Offset: 1

Jud McCranie, Dec 11 1999

nonn

Terms are 0 when Pn == 1 or 4 mod 5, terms are 1 when Pn == 2 or 3 mod 5.

			P3=5, Fibonacci(5-1)=3 mod 5.

Cf. A000045, A003631, A045468, A051830, A051831.

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A086937 Number of distinct zeros of x^2-x-1 mod prime(n).

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A051831 a(n) = Fibonacci(prime(n)) mod prime(n), where prime(n) is the n-th prime.

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A051834 Fibonacci(Pn-1) mod Pn, where Pn is the n-th prime.

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