like atoms In arithmetic, prime numbers always occupy a special place on the number line. The current, Jared Duker Lichtman, a 26-year-old graduate student at Oxford University, solved a famous conjecture, establishing another facet of what makes primes special — and, in a sense, even optimal. “It gives you a broader context to see how primes are unique and in what ways they relate to the larger universe of sets of numbers,” he said.
The conjecture refers to primitive sets — sequences in which no number divides any other number. Since each prime can only be divided by 1 and itself, the set of all primes is an example of the prime. So is the set of all numbers that have exactly two or three or 100 prime factors.
Primitive sets were introduced by mathematician Paul Erdős in the 1930s. At the time, they were simply a tool that made it easy for him to prove something about a certain group of numbers (so-called). is a perfect number) originated in ancient Greece. But they quickly became subjects of interest in their own right — subjects to which Erds would return many times throughout his career.
That’s because, although their definition is simple enough, primitive sets are really weird beasts. That weirdness can be captured by simply asking how large a set of primitives can get. Consider the set of all integers up to 1,000. All numbers between 501 and 1,000 — half of the set — form a primitive set, because no number is divisible by any other number. In this way, primitive sets can cover a large part of the sequence of numbers. But other primitive sets, like the sequence of all primes, are sparse. “It tells you that primitive sets are really a very broad class that you can hardly reach directly,” says Lichtman.
To grasp the interesting properties of sets, mathematicians study different concepts of size. For example, instead of counting how many numbers are in a set, they could do the following: For every number WOMEN in the set, plug it into the expression 1 / (WOMEN log WOMEN), 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 ones, that sum — the “Sum of Erdős” — is always finite. Regardless of what a primitive set looks like, its Erdős sum will always be less than or equal to some number. And so while that sum “looks, at least on the surface, completely foreign and ambiguous,” says Lichtman, it does, in some ways, “control 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 is the maximum possible sum of Erdős. Erdős conjectured that it would be prime, which is about 1.64. Through this prism, the primes form a kind of extreme.
