as an atom In arithmetic, prime numbers always occupy 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, establishing another aspect of prime numbers that are special—in a sense, even optimal. “It gives you a larger context to understand in what ways prime numbers are unique and in what way they relate to the larger set of numbers,” he said.
The conjecture involves primitive sets—sequences in which no numbers divide any other numbers. Since every prime number is only divisible by 1 and itself, the set of all prime numbers is an example of a primitive set. The same goes for the set of all numbers with exactly two or three or 100 prime factors.
Primitive sets were introduced in the 1930s by mathematician Paul Erdős. At the time, they were just a tool to make it easier for him to prove a certain number of ancient Greek origins (called perfect numbers). But they quickly became objects of interest—objects Erdős would return to again and again throughout his career.
That’s because, despite their simple definition, primitive sets are truly strange beasts. This oddity can be captured by simply asking how big the original collection can get. Consider the set of all integers up to 1,000. All numbers from 501 to 1,000 — half of the set — form a primitive set, since no number is divisible by any other number. In this way, the original set may contain a large number of number lines. But other primitive sets, such as the sequence of all prime numbers, are very sparse. “It tells you that the original set is really a very broad category that’s hard to grasp directly,” Lichtman said.
To capture the interesting properties of sets, mathematicians study the concept of various sizes.For example, instead of counting how many numbers are in a set, they might do the following: For each number n In the set, substitute it into the expression 1/(n log n), then add 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 found that for any primitive set, including infinite sets, this sum – “the Erdős sum” – is always finite. Whatever the original set looks like, its sum of Erdős is always less than or equal to some number. So while this sum is “completely alien and vague, at least on the surface,” Lichtman says, it “manages some of the confusion of the original set” in some ways, making it a measuring stick to use correctly.
With this stick in hand, the next natural question to ask is the maximum possible Erdős and how much it could be. Erdős speculated that it would be one of the prime numbers, which turned out to be about 1.64. Through this lens, prime numbers constitute an extreme.
