In the celebration of upcoming Pi Day (3/14, March 14th), this article provides some examples of interesting prime numbers related to the digits of Pi.

Note to the reader: with the exception of the following mathematical definitions no additional knowledge is required for reading this article:

a) A prime number is a natural number greater than 1 which is divisible only by 1 and the number itself and no other number. For example 19 is a prime number (1 x 19) whereas 91 (7 x 13) is not.

b) An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. Example: prime number 31 is an emirp since 13 is a prime number.

Pattern 1: Summation of powers of 314:

Pattern 2: An 867-digit prime number built from 314 surrounded by equal number of 413s

Pattern 3: Prime numbers formed by adding sub-strings composed of Pi’s digits (starting with the first digit)

Pattern 4: An 1191-digit prime number composed of digits 3,1,and 4

Pattern 5: Summation of prime numbers based on Pi’s digits

Statistic: Digit 7 is the most frequent digit in the first digits of 3,141,592

Extra:

a) Albert Einstein was born on 3/14/1879.

b) Einstein’s field equation, relating the curvature of spacetime to energy sources, involves constant Pi:

c) An interesting approximation:

Celebrate Pi Day

a) A prime number is a natural number greater than 1 which is divisible only by 1 and the number itself and no other. Example: 7 is a prime number, but 9 is not.

b) An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. Example: prime number 31 is an emirp since 13 is a prime number.

c) A palindromic number is a number (such as 27172) that remains the same when its digits are reversed.

As an amateur mathematician, I have been interested in exploring/discovering prime numbers whose digits follow some repeating pattern. One of the challenges in doing so is the time complexity involved finding large prime numbers. However, due to rapid advancement and affordability of powerful laptops and cloud-base software, I have been able to explore some of the large primes with interesting patterns in them.

Below is a sample of some of the patterns (with examples). I have grouped them based on the type of repeating pattern in the primes.

1. Pattern: primes composed of consecutive integers:

2. Pattern: primes composed of consecutive digits in numbers pi, phi (Golden Ratio), and e (Euler’s number).

3. Pattern: family of primes, starting with a prime number and embedding zeros in it.

4. Pattern: Palindromic primes:

a) 76…212…67

b) A 1191-digit prime number:

5. Pattern: Family of primes based on sum of the powers of digits in the date string April 9th 2022 (2 0 2 2 4 9)

6. Fibonacci terms that are primes (for the first 500 terms).

Notes:

• the asterisk * denotes a prime number being an emirp as well
• format (for the listed entries below): n : fib(n)

7. Pattern: primes embedded in first 10,000 digits of pi, starting with the first digit:

8. Pattern: a prime number & its emirp composed of consecutive integers (13 through 1) separated by 09

9. Pattern: a prime number composed of consecutive odd integers (1–97)