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

Author

Gary Detlefs, May 04 2012

Keywords

• nonn

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.

Crossrefs

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

Programs

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 *)

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