Chapter 1: The Allure of Prime Numbers

One of the most captivating aspects of mathematics is the study of numbers, particularly prime numbers, which possess distinct qualities that spark curiosity.

Recently, I delved into groundbreaking research from Cornell University regarding prime numbers. Two scholars analyzed the first 100 million prime numbers through sophisticated algorithms, discovering that their distribution is not as random as previously thought. In summary, they concluded that since 2 is the only even prime and numbers ending in 5 (greater than 5) cannot be prime, all primes must end in 1, 3, 7, or 9. While one might expect these last digits to occur evenly, the study revealed that primes ending in 3 and 7 appeared 30% of the time, those ending in 9 had a 22% occurrence, and intriguingly, those ending in 1 only appeared 18.5% of the time. This indicates that prime numbers follow a pattern yet to be fully understood.

Why is this prime number research significant? It’s because cryptography, which safeguards our privacy and information, relies heavily on prime numbers. Understanding their underlying patterns could potentially revolutionize our society.

Previously, I discussed the Riemann Hypothesis, which is closely related to this subject. For more insights, you can check that out below.

Returning to our topic, prime numbers are defined as integers greater than 0 that can only be divided by 1 and themselves. They begin with 2 and continue as 3, 5, 7, 11, 13, and so forth, extending infinitely. This infinite nature of prime numbers was brilliantly proven by Euclid over two millennia ago. For those interested, further details on his method can be found below.

Given the immense fascination surrounding prime numbers, we have discovered countless examples. However, some stand out due to their remarkable properties. Here, I will share what I consider to be the 16 most intriguing prime numbers.

Before diving into my favorite primes, I recommend three enlightening books for those wishing to explore the world of prime numbers:

• The Book of Prime Number Records by Paulo Ribenboim
• Prime Numbers: The Most Mysterious Figures by David Wells
• Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire

Chapter 2: Record-Setting Prime Numbers

The title of the largest known prime number goes to 2⁸²⁵⁸⁹⁹³³ − 1, which boasts 24,862,048 digits. Discovered by Patrick Laroche during the Great Internet Mersenne Prime Search, this number is so large that writing it out would require 13,585 pages—an impossible task for any computer to handle.

In another article, I will explore why mathematicians devote their lives to discovering ever-larger prime numbers.

1. 111111... (1000 digits)

Remarkably, writing the digit 1 a thousand times yields a prime number—a great conversation starter.

2. 1000...2569 (201 digits)

This prime number is fascinating due to its multitude of zeros.

3. 23456789

This is the largest known prime with digits in a sequentially increasing order. One can't help but wonder how impressive it would be if it began with 1.

4. 4567890123... (197 digits)

Writing the digits 4567890123 nineteen times followed by 4567 results in an extraordinary prime.

5. Belphegor's Prime

Discovered by Harvey Dubner, 1000000000000066600000000000001 is known as Belphegor's Prime. Its unique structure—a palindromic form starting and ending with 1—makes it particularly intriguing.

6. 1808010808...1 (15601 digits)

This prime is formed by repeating 1808010808 exactly 1560 times and then adding 1.

7. 73939173

This number is unique as removing the last digit leaves another prime. This pattern continues, with 73939133 being the largest known prime that exhibits this property.

8. 357686312646216567629137

Similar to the previous prime, this one allows for the removal of digits from the start while still remaining prime.

9. 72323252323272325252... (3120 digits)

This prime is remarkable as it consists entirely of prime digits and is formed by repeating the number 72323252323272325252 one hundred fifty-six times.

10. 1226280710981

An ordinary-looking prime that reveals an interesting property when expressed in a certain way.

11. 82818079...1

Start from 82 and write down every integer less than it down to 1, resulting in a prime number.

12. 999...8...999 (506 digits)

This is my favorite prime. By writing the digit 9 505 times and placing an 8 in the middle, you create a prime number.

13. 649-digit prime in e

Writing 651 digits of the number e yields a prime, demonstrating the importance of irrational numbers in mathematics.

14. The first 38 digits of pi

Although shorter than the previous prime, the first 38 digits of pi also form a prime. It’s notable that all known and unknown primes are contained within pi, but we’ve only identified a fraction so far.

15. The Prime Number 2

Lastly, we have the only even prime number, 2, which holds a special place on this list despite being mentioned last.

In this video, "Probably the Most Interesting Prime Number," you’ll discover the unique properties and patterns of prime numbers that make them so fascinating.

"The High Schooler Who Solved a Prime Number Theorem" explores the impact of prime numbers on modern mathematics and cryptography, shedding light on their significance.