as the atoms of arithmetic, prime numbers have always occupied a special place on the number line. Now, Jared Duker Lichtman, a 26-year-old graduate student at the University of Oxford, has solved a well-known conjecture and established another facet of what makes prime numbers special – and in a sense even optimal. “It gives you a larger context to see in what ways prime numbers are unique and in what ways they relate to the larger universe of number sets,” he said.
The conjecture deals with primitive quantities – sequences where no number divides anything else. Since each prime number can only be divided by 1 and itself, the set of all prime numbers is an example of a primitive set. The same is the amount of all numbers that have exactly two or three or 100 prime factors.
Primitive sets were introduced by the mathematician Paul Erdős in the 1930s. Back then, they were simply a tool that made it easier for him to prove something about a certain class of numbers (called perfect numbers) with roots in ancient Greece. But they quickly became the subject of interest in their own right – the ones that Erdős would return to again and again throughout his career.
This is because, although their definition is straightforward enough, primitive sets turned out to be strange beasts. That oddity could be captured by simply asking how big a primitive set can get. Consider the set of all integers up to 1,000. All the numbers from 501 to 1,000 – half of the set – form a primitive set, as no number is divisible by anything else. In this way, primitive sets can form a large part of the number line. But other primitive sets, like the order of all prime numbers, are incredibly sparse. “It tells you that primitive sets are really a very broad class, which is hard to get your fingers in directly,” Lichtman said.
To capture interesting properties of quantities, mathematicians study different notions of size. For example, instead of counting how many numbers are in a set, they can do the following: For each number n in the set, put it in the expression 1 / (n onion n), and then add all the results together. For example, the size of the set {2, 3, 55} becomes 1 / (2 log 2) + 1 / (3 log 3) + 1 / (55 log 55).
Erdős found that for any primitive set, including infinity, this sum – “Erdős sum” – is always finite. No matter what a primitive set may look like, its Erdős sum will always be less than or equal to a number. And so even though that sum “at least on the front page looks completely foreign and vague,” Lichtman said, “it somehow controls some of the chaos of primitive sets,” making it the right dipstick to use.
With this stick in hand, a natural next question is to ask what the maximum possible Erdős sum can be. Erdős assumed that it would be the one for the prime numbers, which comes out to around 1.64. Through this lens, the prime numbers form a kind of extreme.