Pi prime

Short description: Prime number containing consecutive digits of pi

In mathematics, a pi prime is a prime number appearing in the decimal expansion of the constant pi.

Definition

Let ${\pi }_n$ be the integer formed by the first $n$ digits of pi. We call ${\pi }_n$ a pi prime when its value is a prime number. The first four pi primes are as follows:

• ${\pi }_1=3$
• ${\pi }_2=31$
• ${\pi }_6=314159$
• ${\pi }_{38}=31415926535897932384626433832795028841$

An alternative definition only considers the decimal portion of pi, i.e. digits after the starting 3. In this definition we define ${\pi }_k$ to be the integer formed by the first $k$ digits of the decimal part of pi. Using this definition, ${\pi }_k$ is prime when $k=5,12,281,547$.

History

The largest known pi prime with 613,373 digits was discovered by Adrian Bondrescu in 2016. Before that the largest number discovered by Eric Weisstein in 2006 had only 78073 digits. It is probable that the next prime in this sequence will be much harder to find. The expected number of primes does not reach six until $n=561460$ digits.

See also

• Pi
• List of prime numbers

External links

• Weisstein, Eric W.. "Pi-Prime". http://mathworld.wolfram.com/Pi-Prime.html. 
• OEIS sequence A005042 (Primes formed by the initial digits of the decimal expansion of Pi)