A182554 Composite numbers k that divide Fibonacci(k+1) or Fibonacci(k-1).

323, 377, 442, 1891, 2737, 2834, 3827, 4181, 5777, 6479, 6601, 6721, 8149, 10877, 11663, 13201, 13981, 15251, 17119, 17711, 18407, 19043, 20999, 23407, 25877, 27323, 30889, 34561, 34943, 35207, 39203, 40501, 44099, 47519, 50183, 51841, 51983, 52701, 53663

Offset

1,1

Comments

Pseudoprimes to a Fibonacci criterion for primality.

It is known that for prime p <> 5, Fibonacci(p-1) or Fibonacci(p+1) is divisible by p. (see Burton reference)

Primes for which Fibonacci(p-1) are divisible by p are congruent to {0,1,4} mod 5 and are listed in A038872.

Primes for which Fibonacci(p+1) are divisible by p are congruent to {2,3} mod 5 and are listed in A003631.

For n <= 1000, a(n) is squarefree (see A005117). - Dmitry Kamenetsky, Jul 20 2015

Any nonsquarefree term is divisible by the square of a Fibonacci-Wieferich prime (i.e., a prime p such that Fibonacci(j) == 0 (mod p^2) for some j not divisible by p). No Fibonacci-Wieferich primes are known, and there are none < 2*10^14, although it is conjectured that there are infinitely many. - Robert Israel, Jul 22 2015

References

David M. Burton, Elementary Number Theory, Allyn and Bacon, 1980, p. 292, #1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Giovanni Resta)

R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087-2094.

Maple

with(combinat): f:= n-> fibonacci(n): for n from 1 to 40000 do if (f(n+1)/n = floor(f(n+1)/n) or f(n-1)/n=floor(f(n-1)/n)) and not isprime(n) then print(n) fi od;

Mathematica

Select[Range[50000], CompositeQ[#] && (Divisible[Fibonacci[# - 1], #] || Divisible[Fibonacci[# + 1], #]) &] (* Amiram Eldar, Sep 12 2022 *)

Prog

(PARI) p=2; forprime(q=3, 1e5, for(n=p+1, q-1, t=Mod([1, 1; 1, 0], n)^(n-1); if(t[1, 2]==0 || (t*[1, 1; 1, 0]^2)[1, 2]==0, print1(n", "))); p=q) \\ Charles R Greathouse IV, May 05 2012

Crossrefs

Cf. A038872, A003631, A000040, A094395, A005117.

Sequence in context: A340099 A082947 A082948 * A340118 A339517 A217120

Adjacent sequences: A182551 A182552 A182553 * A182555 A182556 A182557

Keyword

nonn

Author

Gary Detlefs, May 04 2012

Status

approved