A phi-prime is a prime number formed by an initial segment of the decimal expansion of the golden
ratio 
,
starting with its integer-part digit 1 and omitting the decimal
point. The first few are 1618033, 1618033988749, ... (OEIS A064117).
The numbers of decimal digits in these examples
are 7, 13, 255, 280, 97241, ... (OEIS A064119).
Rodenkirch (2017) reported that there are no other such primes
with fewer than 
digits.
Another set of phi-related primes is given by 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 ![[phi^n]](/images/equations/Phi-Prime/Inline7.svg)
is prime, where ![[x]](/images/equations/Phi-Prime/Inline8.svg)
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).
(phi)." Jun. 20, 2017.