# A side project of a graduate student proves an assumption for a prime number

such as atoms from arithmetic, prime numbers have always occupied a special place in numerical rights. Now, Jared Ducker Lichtman, A 26-year-old graduate student at Oxford University, has solved a well-known hypothesis by establishing another aspect of what makes prime numbers special – and in a sense even optimal. “This gives you a broader context to see how prime numbers are unique and how 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 number can be divided only by 1 and itself, the set of all primes is an example of a primitive set. The same is the set of all numbers that have exactly two or three or 100 prime factors.

Primitive sets were introduced by mathematician Paul Erdosh in the 1930s. At the time, they were simply an instrument 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 an object of interest in their own right – ones Erdosh will return to again and again throughout his career.

This is because, although their definition is clear enough, primitive sets turned out to be really strange beasts. This oddity can be captured by simply asking how big a primitive set can become. Consider the set of all integers up to 1000. All numbers from 501 to 1000 – half of the set – form a primitive set, since no number is divisible by another. Thus, primitive sets can include a large part of numerical rights. But other primitive sets, such as the sequence of all prime numbers, are incredibly scarce. “This tells you that primitive sets are really a very broad class that is difficult to get directly,” Lichtman said.

To capture the 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 include it in the expression 1 / (n diary n), then sum all the results. The size of the set {2, 3, 55}, for example, becomes 1 / (2 log 2) + 1 / (3 log 3) + 1 / (55 log 55).

Erdosh found that for any primitive set, including infinite ones, this sum – “Erdosh’s sum” – is always finite. No matter what a primitive set may look like, its sum of Erdős will always be less than or equal to some number. So while this amount “seems, at least at first glance, completely foreign and obscure,” Lichtman said, it somehow “controls part of the chaos of primitive sets,” making it the right measuring rod to use.

With this stick in hand, the next natural question you need to ask is what can be the maximum possible amount of Erdosh. Erdosh suggested that this would be for prime numbers, which is about 1.64. Through this lens, prime numbers represent a kind of extreme.