• An odd new paper without peer review claims prime numbers have “genes,” “roots,” and “offspring.”

• Prime numbers are essential to modern life because they underpin all of encryption.

• What is written makes no sense, but there may be good ideas, buried in it, for future discussion.

In new research that is not yet peer evaluated, mathematicians from Hong Kong and North Carolina say they’ve figured out a way to predict prime numbers. But the paper is confusing and makes bizarre claims. If this prime number news is credible, then its authors will need to go back to the drawing board with how they define and explain their concept. The media must stop repeating claims like those in this paper without criticizing them. And in the meantime, prime numbers will continue to rumble through the modern world as the engines of encryption.

A prime number is an integer that can’t be divided by any integer other than itself and 1. In school math class, you probably learned something called prime factorization, where you break a number down into a formula of primes. Let’s say the number is 100: that’s 25 times 4, or 5 times 5 times 2 times 2. This can be a big help when trying to find common factors between numbers. It’s also a building block of number theory.

Large prime numbers are massively useful in cryptography, which is the study and use of codes. Let’s say you’re a hacker who wants to intercept secure messages from one government to another. You can easily write an algorithm that will try to factor the messages’ encryption key, which is just a very large integer. But it takes a lot of time and computing power to try all the prime numbers up to a hundred or more digits long. This is how encryption stays a hair’s breadth ahead of hackers, and it’s one reason why quantum computing, with its suggestion of much more computing power, is such a scary idea to those who write encryption algorithms.

Prime numbers are infinite, and proving anything about patterns in them is notoriously difficult. Mathematicians continue to chip away at pair primes, which are primes that are two apart: 11 and 13, 29 and 31, 617 and 619. But proving anything like this that extends into infinity requires a kind of proof called induction, where you need to turn concrete numbers into an airtight formula that works for all values extending to infinity. It’s like plotting the simple equation y = x, but instead of a line, your formula represents the entire idea of pair primes.

In their paper, The Periodic Table of Primes, researchers Han-Lin Li, Shu-Cherng Fang, and Way Kuo lay out an ill-defined theory that prime numbers, to them, appear to repeat in some way. Way Kuo retired in 2023 as President of City University Hong Kong, and he trained as an engineer, not a mathematician. The others also received their PhDs decades ago, and they primarily do research in algorithms, not pure mathematical number theory.

“We identify 48 integers out of a period of 2 x 3 x 5 x 7 = 210 to be the roots of all primes as well as composites without factors of 2, 3, 5, and 7,” the researchers explain. They can’t mean “roots” literally, as, by definition, no prime number has an integer square root or any other factor besides itself and 1. As soon as you multiply any of these integers together, you get a composite number, not a prime.

“Each prime, twin primes, or composite without factors of 2, 3, 5 and 7 is an offspring of the 48 integers uniquely allocated on the [Periodic Table of Primes],” they continue. What is an offspring? A multiplied product? Again, that’s not possible. And their “formula” is based on the idea that two prime numbers multiplied together forms a composite integer. “For a long time, many said these integer roots grew like weeds among natural numbers, but nobody could predict where the next primes may sprout,” they write. What does that mean?

This paper could be an April Fools joke, but it was uploaded more than two weeks ahead on March 15. If there’s validity in these claims, it will come with clearer definitions, strong mathematical language, and the support of peer reviewers who are active in math research.

You Might Also Like