OFFSET
1,1
COMMENTS
Fibonacci numbers grow exponentially while the average gap between consecutive primes grows logarithmically, thus this sequence's density decreases as k increases.
It is conjectured that this sequence is infinite as a consequence of the Polignac's conjecture.
Infinite if the Twin Prime conjecture holds. - Michael S. Branicky, May 24 2026
LINKS
Vincenzo Manto, Table of n, a(n) for n = 1..1000
EXAMPLE
7 (4th prime) is in the sequence because it follows 5 (3rd prime), and 7 - 5 = 2 in A000045.
61 (18th prime) is in the sequence because it follows 59 (17th prime), and 61 - 59 = 2 is in A000045.
11 (5th prime) is not in the sequence because it follows 7 (4th prime), and 11 - 7 = 4, is not in A000045.
MAPLE
q:= p-> (t-> ormap(issqr, [t-4, t+4]))(5*(p-prevprime(p))^2):
select(isprime and q, [$3..1055])[]; # Alois P. Heinz, May 23 2026
MATHEMATICA
With[{p = Prime[Range[177]]}, p[[1 + Position[Differences[p], _?(Or @@ IntegerQ /@ Sqrt[5*#^2 + {-4, 4}] &), 1] // Flatten]]] (* Amiram Eldar, May 24 2026 *)
PROG
(Python)
from math import isqrt
from sympy import prime
def is_fibonacci(n):
def is_sq(x):
s = isqrt(x)
return s*s == x
return is_sq(5*n*n + 4) or is_sq(5*n*n - 4)
def sequence(n_terms):
results = []
for i in range(2, n_terms + 1):
gap = prime(i) - prime(i-1)
if is_fibonacci(gap):
results.append(prime(i))
return results
print(sequence(150))
CROSSREFS
KEYWORD
nonn
AUTHOR
Vincenzo Manto, May 23 2026
STATUS
approved