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While I was looking for a gift for a child’s birthday, a math book fell into my hands. I am always fascinated when authors write about abstract scientific topics for children, whether it’s on Albert Einstein’s theories, the life of Marie Curie, technology, or space travel. But this particular book was different. It’s all about prime numbers—specifically twin primes. Danish author Jan Egesborg has endeavored to introduce children to one of the most stubborn open problems in number theory, which even the brightest minds have repeatedly failed to solve over the past 100-plus years: the twin prime conjecture.

As is so often the case in mathematics, the conjecture falls into the category of those that are easy to understand but devilishly hard to prove. Twin primes are two prime numbers that have a distance of two on the number line; that is, they are directly consecutive if you ignore even numbers. Examples include 3 and 5, 5 and 7, and 17 and 19. You can find a lot of twin primes among small numbers, but the farther up the number line you go, the rarer they become.

That’s no surprise, given that prime numbers are increasingly rare among large numbers. Nevertheless, people have known since ancient times that infinite prime numbers exist, and the prime number twin conjecture states that there are an infinite number of prime number twins, as well. That would mean that no matter how large the values considered, there will always be prime numbers in direct succession among the odd numbers.

Admittedly, translating these concepts for kids is not easy (which is why I have so much respect for Egesborg and his children’s book). Prime numbers (2, 3, 5, 7, 11, 13,…) are like the fundamental particles of the natural numbers. They are only divisible by 1 and themselves. All other natural numbers can be broken down into their prime divisors, which makes prime numbers the basic building blocks of the mathematical world.

A Proof from Antiquity

Mathematics has an unlimited number of prime number building blocks. Euclid proved this more than 2,000 years ago with a simple thought experiment. Suppose there were only a finite number of prime numbers, the largest being p. In this case, all prime numbers up to p could be multiplied together.

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