Meet the largest known prime: it’s got 41 million digits and was calculated using GPUs

As of October 2024, we have found the biggest prime number we know of — and it took almost 6 years to find it. To generate this number, you multiply 2 by itself 136,279,841 times (resulting in an enormous number that’s almost impossible to fathom) and then subtract 1. Luke Durant, a 36-year-old researcher and former NVIDIA employee, is the one who found the number.

The new big prime on the block

The newly discovered prime number, referred to as M136279841, is a Mersenne prime. This is a special type of prime number named after the 17th-century French mathematician Marin Mersenne. Mersenne primes are calculated using the formula 2ⁿ-1, meaning that the number is generated by multiplying two by itself n times, and then subtracting one from the result.

With 41,024,320 digits, this prime is the largest known to date, eclipsing the previous record by more than 16 million digits. The discovery was made on October 11, 2024, when an Nvidia A100 GPU in Dublin, Ireland, produced the probable prime. Its primality was confirmed a day later by an Nvidia H100 GPU in San Antonio, Texas, using the Lucas-Lehmer test, a powerful algorithm designed to verify Mersenne primes. This ends the 28-year reign of ordinary personal computers finding these huge prime numbers.

The effort to find this prime number has been spearheaded by Great Internet Mersenne Prime Search (GIMPS). The project relies on distributed computing, where volunteers from around the world contribute their computing power to test candidate numbers for primality. Traditionally, this was done using personal computers, but as numbers grew larger, so did the need for more powerful hardware.

In 2017, programmer Mihai Preda developed GpuOwl, a software tool that allows GIMPS users to test Mersenne numbers on GPUs (graphics processing units). This was a game-changer for the project, as GPUs significantly sped up the search for large primes. When Luke Durant joined GIMPS in 2023, he saw the potential of using cloud-based GPU servers to take this effort to the next level. By developing infrastructure that could deploy GpuOwl across thousands of GPUs in data centers around the world, Durant dramatically increased the computational power available to GIMPS.

The discovery of M136279841 is the first prime found using this new GPU-driven approach, and it signals a new era for both prime hunting and the broader use of GPUs in scientific research. As Durant put it, “GPUs aren’t just for AI and gaming. They’re incredibly versatile and can be used for everything from cryptography to large-scale simulations.”

What’s the deal with prime numbers

Prime numbers are natural numbers greater than one that have no divisors other than 1 and themselves. In other words, a prime number cannot be formed by multiplying two smaller natural numbers. For example, 2, 3, 5, and 7 are prime numbers because they can only be divided evenly by 1 and the number itself, while 4 and 6 are not prime because they can be divided by smaller numbers like 2 or 3.

Prime numbers are considered the building blocks of all whole numbers, as any number greater than one can be factored into a product of primes, a concept known as the Fundamental Theorem of Arithmetic. This unique property gives primes a central role in number theory and has led to their use in fields like cryptography, where large primes help secure data through encryption methods.

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Despite their simplicity, prime numbers are fascinating because of their unpredictable distribution and the unsolved mysteries surrounding their patterns.

Mersenne primes are somewhat easier to find compared to other large prime numbers because they follow a specific mathematical form, 2^n-1, which allows for efficient testing using specialized algorithms. Additionally, the predictable pattern of Mersenne primes reduces the number of candidates that need to be tested, focusing the search and making it more manageable, especially with the help of powerful computers and distributed computing projects like GIMPS.

There are 52 Mersenne prime numbers and they have been the main focus of the study of the largest prime numbers. However, there are many unknowns even about these numbers. It is not even known whether the set of Mersenne primes is finite or infinite.

How big this number is

It’s hard to wrap your mind around a number like 2^136,279,841-1 but let’s try.

Here’s how the powers of 2 work:

Let’s look at some examples:

• 2² means doubling 2 once, so 2 times 2 equals 4.
• 2³ means doubling 2 twice, so 2 times 2 times 2 equals 8.
• 2⁴ means doubling 2 three times, so 2 times 2 times 2 times 2 equals 16.

Let’s fast-forward a bit:

• 2¹⁰ is 1024;
• 2²⁰ is 1048576;
• 2³⁰ is 1,073,741,824;
• 2¹⁰⁰ is 1,267,650,600,228,229,401,496,703,205,376.

It’s not hard to see how fast these numbers get really big.

Now, imagine doing this 136,279,841 times. That’s what 2^136,279,841 means — a super huge number created by doubling 2 over 136 million times. After we do that, we subtract 1. This gives us the biggest known prime numbers, which is so large it has more than 41 million digits.

Why do we search for very large prime numbers?

In a direct sense, the search for large prime numbers might seem like a purely academic pursuit. After all, what practical use could a 41-million-digit number possibly have? It’s true that these giant primes don’t have immediate applications, at least for now — but their discovery can have ripple effects across multiple fields of science and technology.

Prime numbers, especially large ones (but not this large), play a key role in cryptography, the science of encoding and decoding information. Modern encryption systems rely on the fact that it’s extremely difficult to factor large numbers into primes. As encryption becomes more sophisticated, larger primes could be used to create even more secure systems.

Moreover, the algorithms and computational techniques developed to find these primes can often be applied to other areas of science. The same GPUs that are used to test prime numbers can also be used for tasks like climate modeling, protein folding, and even developing new materials.

At its core, the search for large primes is about pushing the boundaries of what we know and what we can do with technology. And with each new discovery, we come one step closer to understanding the mysteries of mathematics and the universe itself.