Pi prime

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 this value is a prime number. The first four pi primes are

• $\pi _{1}=3$
• $\pi _{2}=31$
• $\pi _{6}=314159$
• $\pi _{38}=31415926535897932384626433832795028841$.

In 2026, Robert Baillie proved that $\pi _{16208}=3141592653...7943936307$ is the fifth pi prime.

History

In 1979, Robert Baillie and Marvin Wunderlich found that $\pi _{38}$ is prime.

In 2001, Ed T. Prothro found that the first 16208 digits of pi form a probable prime, and that after $\pi _{38}$, all such numbers with fewer than 16208 digits are composite numbers.

In 2006, Eric Weisstein showed that the next two possible pi primes have 47577 and 78073 digits. In 2016, A. Bondrescu showed that, after 78073, the next possible pi prime has 613373 digits. As of 2026, it has not been proven that any of these three numbers are primes.

A heuristic argument suggests that the sequence of pi primes is infinite.

Alternative definitions

One alternative definition considers only 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 $k=5,12,281,547$.

See also

• Pi
• List of prime numbers

External links

• Weisstein, Eric W. "Pi-Prime". MathWorld.
• OEIS sequence A005042 (Primes formed by the initial digits of the decimal expansion of Pi)