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Pi-Prime


A pi-prime is a prime number appearing in the decimal expansion of pi. The known examples are 3, 31, 314159, 31415926535897932384626433832795028841, ... (OEIS A005042). The numbers of digits in these examples are 1, 2, 6, 38, 16208, 47577, 78073, 613373, ... (OEIS A060421). The largest of these are summarized in the following table.

decimal digitsdiscovererdate
47577E. W. WeissteinApr. 1, 2006
78073E. W. WeissteinJul. 13, 2006
613373A. BondrescuMay 29, 2016

Another set of pi-related primes is the positive integers n such that |_pi^n_| is prime, where |_x_| is the floor function. The first few are 1, 3, 4, 12, 73, 317, 2728, 6826, 7683, 7950, 14417, ... (OEIS A059792), corresponding to the primes 3, 31, 97, 924269, ... (OEIS A077547).

Similarly, the first few n such that [pi^n] is prime, where [x] is the ceiling function are 5, 29, 88, 948, 1071, 1100, 1578, ... (OEIS A111937) with no others less than 10^4, corresponding to the primes 307, 261424513284461, 56129192858827520816193436882886842322337671, ... (OEIS A118843).


See also

e-Prime, Constant Primes, Integer Sequence Primes, Phi-Prime, Pi

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References

Brown, K. S. "Primes in the Decimal Expansion of Pi." http://www.sixfingeredman.net/ref/mathpages-notes/kmath184/kmath184.htm.Prime Curios! "314159." http://primes.utm.edu/curios/page.php?short=314159.Sloane, N. J. A. Sequences A005042/M3129, A059792, A077547, A060421, A111937, and A118843 in "The On-Line Encyclopedia of Integer Sequences."

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Pi-Prime

Cite this as:

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

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