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Phi-Prime


A phi-prime is a prime number appearing in the decimal expansion of the golden ratio phi. 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 500000 digits (M. Rodenkirch, Jun. 20, 2017).

Another set of phi-related primes is the positive integers n such that |_phi^n_| is prime, where |_x_| 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 n such that [phi^n] is prime, where [x] is the ceiling function are 1, 2, 3, 4, 8, 16, ... (OEIS A118841), with no others less than 10^5, corresponding to the primes 2, 3, 5, 7, 47, 2207, ... (OEIS A118842).


See also

Constant Primes, e-Prime, Golden Ratio, Integer Sequence Primes, Pi-Prime

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References

Rodenkirch, M. "Primes in phi (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."

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Phi-Prime

Cite this as:

Weisstein, Eric W. "Phi-Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Phi-Prime.html

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