3867632931 × 10^10001 +1 - Numberphile

The Discovery of the Largest Known Reversible Prime

Key Concepts:

• Prime Number: A whole number greater than 1 that has only two divisors: 1 and itself.
• Emurp/Reversible Prime: A prime number that remains prime when its digits are reversed. (e.g., 13 and 31 are both prime).
• Palindrome Prime: A prime number that remains the same when its digits are reversed (e.g., 101). Distinguished from Emurps.
• Prime Sieve (Siv): An algorithm for finding prime numbers by iteratively marking multiples of primes as composite.
• PrimeGrid: A distributed computing project dedicated to searching for prime numbers and related mathematical objects.
• Computational Complexity: The amount of resources (time, memory) required to run an algorithm. C++ is noted as being more efficient than Python for this task.

Introduction to Reversible Primes & the New Discovery

The video details the recent discovery of a new record-breaking reversible prime – a prime number that remains prime when its digits are reversed. This prime is unique not just for its size, but for being a specific type of prime known as an Emurp (or reversible prime). The newly discovered prime has 10,001 digits, surpassing the previous record of 9,997 digits by four digits. The presenter emphasizes the rarity of discovering new primes, particularly of this specific type. The example of 121 is given as a simple illustration: 121 is prime, and reversing the digits yields 121, which is also prime, making it an Emurp. Other examples include 1193 and 3911. The presenter clarifies that palindromic primes (like 101) are distinct from Emurps, as they read the same forwards and backwards, while Emurps must change when reversed but remain prime.

The Record-Breaking Prime: Structure and Calculation

The newly discovered prime is constructed using a specific formula: 3867632931 * 10¹⁰⁰⁰¹ + 1. This translates to the number 3,867,632,931 followed by 10,000 zeros, and then a 1. To verify it’s a reversible prime, the entire number is reversed, resulting in a 10,011-digit number. The reversed number is 1 followed by 10,000 zeros, then 1392367683. The presenter humorously initially claimed to have memorized the number, then clarified he could memorize it, and then demonstrated the structure to make it easier to understand.

Methodology & Computational Process

The discovery was made by Stefan, who utilized the distributed computing project PrimeGrid. Stefan developed his own prime sieve (in C++) – described as “terrible” by the presenter, but effective – alongside incorporating existing, optimized prime algorithms available on PrimeGrid. The process involved searching for combinations of numbers, lengths of zeros, and adding a one to the end, then testing if the resulting number and its reverse were both prime. This process requires significant computational power and time. The presenter notes that the efficiency of C++ was crucial for this task, contrasting it with Python. The discovery was initially shared with the presenter via email, a common occurrence given his involvement in the prime number community.

The Role of PrimeGrid & Collaborative Computing

PrimeGrid is highlighted as a crucial platform for this discovery. It’s a collaborative project where individuals contribute computing power to search for prime numbers. The presenter previously ran his own server, “Calculus Prime,” for PrimeGrid, but it was decommissioned due to age and limited processing power. The success of Stefan’s discovery demonstrates the power of combining individual effort with the resources and algorithms developed by the broader PrimeGrid community – a “lone warrior” benefiting from the “shoulders of giants.”

Is This the Last Reversible Prime?

The video raises the question of whether this is the last possible reversible prime. According to Wikipedia, whether an infinite number of Emurps exist is an open problem in mathematics. The presenter speculates that this could be the final Emurp, a “top of the tree” discovery. He acknowledges the infinite nature of prime numbers but suggests that finding increasingly larger Emurps may become increasingly difficult. He uses an orchard analogy: while there's an infinite amount of fruit, some trees are easier to harvest than others. Finding Emurps represents harvesting from a less-accessible tree.

The Computer Behind the Discovery

The presenter showcases the computer used to find the prime – a small, eight-core machine located in his office. This emphasizes that significant mathematical discoveries don’t always require massive supercomputers; dedicated enthusiasts with accessible hardware can still make substantial contributions.

Brilliant.org Sponsorship & Importance of Foundational Skills

The video includes a sponsored segment for Brilliant.org, an online learning platform. The presenter emphasizes the importance of foundational skills in mathematics and coding for making these kinds of discoveries. He notes that understanding the underlying principles is crucial, and Brilliant.org provides interactive courses to build those skills.

Conclusion & Final Thoughts

The discovery of this new reversible prime is presented as a noteworthy event, not just for its size, but for the combination of individual enthusiasm, collaborative computing, and efficient coding techniques that made it possible. The presenter highlights the fun and rewarding nature of exploring these “freakish cool things in mathematics” and encourages viewers to develop the skills necessary to contribute to future discoveries. He concludes by reiterating the open question of whether more reversible primes exist, leaving the audience with a sense of wonder and the possibility of further exploration.