A phi-prime is a prime number appearing in the decimal expansion of the golden ratio . The first few are 1618033, 1618033988749, ... (OEIS A064117). The numbers of decimal digits in these examples are 7, 13, 255, 280, 97241, ... (OEIS A064119). There are no others with less than  digits (M. Rodenkirch, Jun. 20, 2017).

Another set of phi-related primes is the positive integers  such that  is prime, where  is the floor function. The first few are 2, 5, 6, 7, 11, 13, 17, 19, 24, 31, 37, 41, 47, 48, 53, 61, 71, 79, 96, 113, 313, 353, 503, 613, 617, 863, ... (OEIS A059791), corresponding to the primes 2, 11, 17, 29, 199, 521, 3571, 9349, ... (OEIS A118839).

Similarly, the first few  such that  is prime, where  is the ceiling function are 1, 2, 3, 4, 8, 16, ... (OEIS A118841), with no others less than , corresponding to the primes 2, 3, 5, 7, 47, 2207, ... (OEIS A118842).

REFERENCES:

Rodenkirch, M. "Primes in  (phi)." Jun. 20, 2017. http://www.mersenneforum.org/showthread.php?p=461643#post461643.

Prime Curios! "16180...05887 (280-digits)." http://primes.utm.edu/curios/page.php?number_id=1206.

Sloane, N. J. A. Sequences A064117, A064119, A118839, A118841, and A118842 in "The On-Line Encyclopedia of Integer Sequences."