A396340

Created: 5/28/2026 | Author: Vincenzo Manto , May 23 2026

Primes prime(k) such that prime(k) - prime(k-1) is a Fibonacci number.

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

Sequence Chart

Data

3,5,7,13,19,31,43,61,73,97,103,109,139,151,181,193,199,229,241,271,283,313,349,367,397,409,421,433,457,463,487,499,523,571,601,619,643,661,691,709,727,751,769,811,823,829,859,883,919,937,991,1021,1033,1051

Computational Implementations

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))

Mathematica