Here’s what you’ll learn when you read this story:
Prime numbers are essential for technologies like RSA encryption, which rely on the difficulty of guessing these numerals.
A new paper shows that another area of mathematics called integer partition unlocks “infinitely many new ways” to detect them beyond divisibility.
The team said that this breakthrough arrived by using decades-old methods to answer mathematical questions no one else thought to ask.
Although prime numbers are a mathematical concept everyone learns about in elementary school, extremely large prime numbers form the backbone of some of the most complex technologies in modern society—especially in the realm of cryptography. But in the burgeoning era of quantum computers, which can solve problems exponentially faster than standard computers (including supercomputers), there’s a chance that this kind of previously uncrackable protection could suddenly become very vulnerable.
This has pushed mathematicians—including Ken Ono from the University of Virginia—to continue exploring the frontiers of prime numbers. In September of last year, Ono (along with co-authors William Craig and Jan-Willem van Ittersum) published a paper in the journal Proceedings of the National Academy of Sciences (PNAS) exploring how to find new prime numbers with a novel approach centered around what are called integer partitions. His groundbreaking work scored him recognition for the Cozzarelli Award for originality and creativity, but to understand it, we’ll need to take a few steps back.
A prime number (as you likely know) is an integer that is not divisible by any number other than 1 and itself. While there are technically infinite prime numbers, it’s difficult to find new ones, as they appear in a number line with no pattern. (Currently, the largest known prime number is more than 41 million digits long.) But Ono and his co-authors discovered a connection between prime numbers and integer partitions, which divvy up numbers into all their possible smaller sums—the number four, for example, can be described as 4, as 3 + 1, as 2 + 2, as 2 + 1 + 1, and as 1 + 1 + 1 + 1.
“The prime numbers, the building blocks of multiplicative number theory, are the solutions of infinitely many special ‘Diophantine equations’ in well-studied partition functions,” the authors wrote. “In other words, integer partitions detect the primes in infinitely many natural ways.”
Named for the third-century mathematician Diophantus of Alexandria, these equations can be incredibly complex, but if the resulting answer turns out to be true, that means you’re working with a prime number. This essentially devises a new way to investigate prime numbers that has never been explored before.
“This paper, as excited as I am about it, represents theoretical math that could’ve been done decades ago,” Ono said in a video interview accompanying a press statement. “What I like about our theorem is that if there was a time machine, I could go back to 1950, explain what we done, and it would generate the same level of excitement […] and the experts at that time would understand what we did.”
Ono is intimately familiar with the security implications of prime number research, as he serves on the advisory board for the National Security Agency (NSA). Technologies like RSA encryption rely on the difficulty of detecting prime numbers to safeguard the world’s most sensitive information, so understanding prime numbers from every conceivable angle will be helpful when quantum computers make ferreting out these unfathomably large numbers easier. Speaking with Scientific American, many mathematicians say this work serves as the foundation of a new way of seeing what other mathematical connections can be made using partition functions.
Prime numbers may be elementary, but they remain a fixture of our complex technological future.
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