What is a Prime Number: The Perspective of Numbers as an "Indivisible Way of Life"

── The day the largest prime number in history, with 40 million digits, was discovered

Recently, the largest prime number in history was discovered. It is about 40 million digits long. Since the previous record was 25 million digits, this is a massive jump of 15 million digits all at once. Units like "kei" and "gai" that we learned about in childhood are at most $10^{68}$, which is only about 68 digits long.

Even if you were to write out those digits at 2,500 per page, 40 million digits would amount to 16,000 sheets of manuscript paper. This staggering number took the world by storm based on a single property alone: that it cannot be divided any further.

Why are people so drawn to prime numbers? It is conjectured that prime numbers exist infinitely. However, no matter how far we go, the "largest prime number" has still not been found—or rather, since they continue infinitely in the first place, the concept of a maximum does not exist.

Many people say that this very endlessness is what makes them romantic. But I look at prime numbers from yet another angle. A prime number is a number that can only maintain its shape precisely because it cannot be divided evenly.

A prime number is a number that cannot be divided any further.

The definition of a prime number is simple: a number that has no divisors other than 1 and itself. 6 is not a prime number because it is divisible by 2 and 3. 5 is a prime number because it can only be divided by 5 and 1. 11, 13, and 17 are the same. For small numbers, you can intuitively tell whether they are prime. However, the larger the number of digits, the more determining whether a number is prime becomes a massive undertaking that requires throwing all available computational resources at it.

The number discovered this time is 2 to the 136,278,401st power minus 1. Numbers in this "2 to the Nth power minus 1" form are called Mersenne numbers. This is a type of number discovered by the 17th-century French mathematician Mersenne, and among these, only the ones that are prime numbers are called Mersenne primes.

Realizing that it is more efficient to search for prime numbers among Mersenne numbers rather than at random, humankind has continuously mined this type of number ever since.

Mersenne primes and perfect numbers: The indivisible shapes the world

The group responsible for this discovery is GIMPS (Great Internet Mersenne Prime Search). By connecting computers around the globe into a network and operating them as a simulated giant supercomputer, they continuously search for Mersenne primes. Thousands of computers across 24 centers in 17 countries spend years chasing a single number. Since the prize money awarded to the discoverer is said to be around 400,000 yen, this is not a matter of economic rationality.

Prime numbers also have unexpected practical utility. RSA encryption, a digital communication security technology we use daily, utilizes the difficulty of prime factorization of large numbers. The pursuit of finding prime numbers itself, much like rocket development, fosters technology in places separate from its direct purpose and translates it to society. This is a structure where byproducts are born outside the intended goal.

Even more interesting is the relationship between Mersenne primes and "perfect numbers." A perfect number is a number that is equal to the sum of all its divisors excluding itself. For 6, it is 1+2+3=6; for 28, it is 1+2+4+7+14=28. These perfect numbers can be created from Mersenne primes. When "2 to the N-th power minus 1" is a prime number, multiplying it by 2 to the N-1-th power will always result in a perfect number.

In other words, the discovery of a 40-million-digit Mersenne prime this time also means that the 52nd perfect number has been found. It turns out that while chasing an indivisible number, we suddenly stumbled upon a "number that, when divided, comes right back to itself."

The romance of chasing the "inexpressible"

The biggest mystery in prime number research is that a complete formula to reliably produce prime numbers simply by calculating it has still not been found. The genius Euler discovered the formula N²−N+41. Remarkably, when substituting integers from 1 to 40 for N, they all become prime numbers. However, even that breaks down at 41.

Fermat primes, repunit primes (numbers consisting only of 1s, such as 11), and Bellprime numbers (also known as the beast's number, with 13 zeros on either side of 666)—for hundreds of years, mathematicians have named, classified, and searched for regularities in special types of prime numbers.

About 2,000 years ago, Euclid of ancient Greece already proved that prime numbers are infinite. Even with 2,000 years of time and the full power of modern computers mobilized, the complete picture of prime numbers still cannot be grasped. Yet humanity continues to chase things that cannot be fully expressed or captured by rules. This is one of the true natures of romance.

People live as "composite numbers"

So, moving away from specialized mathematical topics, from here on I would like to take a slightly more imaginative perspective. Most humans live as "composite numbers" rather than prime numbers. A composite number is a number made by multiplying multiple elements together: 6 is 2×3, 10 is 2×5. People too are made of who they are today through the multiplication of various "divisors"—the environment they grew up in, the people they met, the influences they received, and the communities they belong to.

Having many divisors also means being easily relatable. If you and someone else share a common divisor, people can connect through that. Living like a composite number means having many parts within yourself that overlap with others. This in itself is neither a bad thing nor inferior. The majority of society is built upon these overlaps.

Living as a prime number──Being indivisible is not loneliness, but strength.る

On the other hand, there are a tiny handful of people who simply cannot be divided evenly. At a glance, they seem large, complex, and packed with various elements, yet in reality, they cannot be broken down by anything other than their own core axis. This is the "prime number" way of living. The larger the number, the more divisible elements it seems to have. Yet prime numbers defy that expectation and cannot be divided. The fact that they aren't even even numbers—all prime numbers greater than two are odd—is symbolic. They lack from the very beginning even the quality that would most naturally make them divisible.

People who live like prime numbers are hard to relate to. This is because they have few divisors within themselves that they can share with others. That is precisely why they look lonely. However, that is not a deficiency, but loneliness as strength. Without being influenced too much by anyone else, they move forward in the direction they want to go and do what they want to do based solely on their own axis. Having few divisors is both the difficulty of being empathized with and the strength of not being easily broken down by others.

And while there is no superiority or inferiority in being a prime number, magnitude has meaning. If I am to live the same indivisible way of life, I want to be a larger prime number. Even if 40 million digits is the maximum today, a prime number with 80 million digits might be found in a few years. At that time, I want to become a presence that makes people wonder in astonishment, "Is someone this indivisible existing on such a grand scale?" Living as a prime number may not be about comparison or competition, but rather about growing gradually larger while remaining indivisible.

The day has not yet come when prime numbers can be completely explained by mathematical formulas. In the same way, people who live lives that cannot be neatly divided cannot be fully explained by existing theories and categories. And that is fine. The elusiveness of explanationRemaining in one's own axis without changing size. There is certainly a perspective that can only be found there.

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