Mathematicians Make a Major Discovery About Prime Numbers | WIRED

December 23, 2014

Major Discovery About #Prime Numbers Extension of trick to find spans of composites, eg start w/ 101! add 2,3,4…101

The two new proofs of Erdős’ conjecture are both based on a simple way to construct large prime gaps. A large prime gap is the same thing as a long list of non-prime, or “composite,” numbers between two prime numbers. Here’s one easy way to construct a list of, say, 100 composite numbers in a row: Start with the numbers 2, 3, 4, … , 101, and add to each of these the number 101 factorial (the product of the first 101 numbers, written 101!). The list then becomes 101! + 2, 101! + 3, 101! + 4, … , 101! + 101. Since 101! is divisible by all the numbers from 2 to 101, each of the numbers in the new list is composite: 101! + 2 is divisible by 2, 101! + 3 is divisible by 3, and so on. “All the proofs about large prime gaps use only slight variations on this high school construction,” said James Maynard of Oxford, who wrote the second of the two papers.