A Grad Student’s Side Project Proves a Prime Number Conjecture

Prime numbers have always been on the number line. A graduate student at the University of Oxford has solved a well known question about what makes the primes special. He said that it gave him a bigger context to see how the primes relate to the larger universe of numbers.

primitive sets aresequences in which no number divides any other One example of a primitive set is the set of all prime numbers. The set of numbers have at least two or three prime factors.

The first primitive sets were created by the mathematician Paul Erds. He was able to prove something about a certain class of numbers with roots in ancient Greece by using them. They became objects of interest to Erds throughout his career.

Their definition is simple, but primitive sets turned out to be weird beasts. Asking how big a primitive set can get is enough to capture strangeness. Consider the number up to 1000. Half of the set is made up of numbers from 500 to 1000. A lot of the number line could be comprised of primitive sets. The sequence of all primes is extremely sparse. The fact that primitive sets are hard to get your hands on directly tells you that.

Calculating the size of sets is one of the things mathematicians do. Plug the number into the expression 1/(n log n) and add up the results. For instance, the set 2, 3, 55 becomes 1/(2 log 2) + 1/ (3 log 3) + 1/ (55 log 55)

The "Erds sum" is finite for any primitive set. The sum of the set's Erds will always be less than or equal to something. The sum looks alien and vague, but it is the right measuring stick to use because of how chaotic it is.

With this stick in hand, it's a good idea to ask what the maximum possible Erds sum might be. The prime numbers come out to about 1.64. The primes are considered to be a kind of extreme.