A pi-prime is a prime number formed by an initial segment of the decimal expansion of pi, including the initial 3 and omitting the decimal point. The known examples are 3, 31, 314159, 31415926535897932384626433832795028841, ... (OEIS A005042). Including reported probable primes, the numbers of digits are 1, 2, 6, 38, 16208, 47577, 78073, 613373, ... (OEIS A060421). Prothro (2001) found the 16208-digit example as a probable prime on Dec. 13, 2001 and R. Baillie has reported proving this number prime using PRIMO and elliptic curve primality proving (R. Baillie, pers. comm., May 12, 2026). The 47577-, 78073-, and 613373-digit examples were reported as probable primes (Havermann 2018). The following table summarizes the largest reported examples.
| decimal digits | discoverer | date | links |
| 16208 | E. T. Prothro | Dec. 13, 2001 | PrimePages, FactorDB |
| 47577 | E. W. Weisstein | Apr. 1, 2006 | |
| 78073 | E. W. Weisstein | Jul. 13, 2006 | |
| 613373 | A. Bondrescu | May 29, 2016 |
Another set of pi-related primes is the positive integers 
such that 
is prime, where 
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 positive integers 
such that ![[pi^n]](/images/equations/Pi-Prime/Inline5.svg)
is prime, where ![[x]](/images/equations/Pi-Prime/Inline6.svg)
is the ceiling
function, are 5, 29, 88, 948, 1071, 1100, 1578, ... (OEIS A111937)
with no others less than 
,
corresponding to the primes 307, 261424513284461,
56129192858827520816193436882886842322337671, ... (OEIS A118843).
