as the atoms of arithmetic, prime numbers have always occupied a special place on the number line. utilities, Jared Duker Lichtman, a 26-year-old graduate student at the University of Oxford, has solved a well-known conjecture and identified another facet of what makes primes special — and in a sense, optimal. “It gives you greater context to see the ways the prime numbers are unique and the ways they relate to the larger universe of sets of numbers,” he said.
The conjecture deals with primitive sets – sequences in which no number divides another. Since any prime can only be divided by 1 and itself, the set of all primes is an example of a primitive set. This also applies to the set of all numbers that have exactly two or three or 100 prime factors.
Primitive sets were introduced in the 1930s by the mathematician Paul Erdős. At the time, they were just a tool that made it easier for him to prove something about a particular class of numbers (so-called perfect numbers) with roots in ancient Greece. But they soon became objects of interest in their own right, which Erdős would return to time and again throughout his career.
That’s because, while their definition is simple enough, primitive sets turned out to be strange beasts indeed. That strangeness can be captured by simply asking how big a primitive set can get. Consider the set of all integers up to 1,000. All numbers from 501 to 1,000 – half the set – are a primitive set, since no number is divisible by another. In this way, primitive sets could make up quite a bit of the number line. But other primitive sets, such as the order of all primes, are incredibly scarce. “It tells you that primitive sets are really a very broad class that is hard to get your hands on right away,” Lichtman said.
To capture interesting properties of sets, 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, plug it into the expression 1/(n log n), then add up all the results. For example, the size of the set {2, 3, 55} becomes 1/(2 log 2) + 1/(3 log 3) + 1/(55 log 55).
Erdős discovered that for any primitive set, including infinite ones, that sum – the “Erdős sum” – is always finite. Whatever a primitive set looks like, the Erdős sum will always be less than or equal to a certain number. And while that sum “looks completely strange and vague at first glance,” Lichtman said, “it kind of checks some of the chaos of primitive sets,” making it the right measure to use.
With this stick in hand, a natural next question to ask is what the maximum possible Erdős sum could be. Erdős guessed it would be the one for the primes, which comes out at about 1.64. Because of this lens, the primes form a kind of extreme.
