Some progress was made towards a proof over the years. The conjecture was true for certain types of primitive sets.
Greg Martin is a mathematician at the University of British Columbia who has worked on related problems. Andrs Srkzy is a mathematician at Etvs Lornd University. He said that it seemed far off.
The primitive set conjecture was started by Lichtman during his final year of college. I was very interested in this question. He said it was very strange how something like this could be true. It has been my companion for the last four years.
He and Carl Pomerance came out of retirement to work with him. Martin said it was not too far away. About 10% larger than the primes.
The constant was obtained by linking a new sequence of multiples to a set of numbers. The primitive set is 2, 3,55. The sequence of all numbers is associated to the number 2. Multiples of 3 are not the same as multiples of 2 The number 55 is associated to multiples of 55 with the smallest prime factor being 11. It's similar to how a dictionary organizes words by using primes rather than letters.
They thought about how much of the number line they occupied was dense. Half of all numbers have a density of 1/2, and the sequence of all even numbers has a density of 1/2 as well. If the original set was primitive, the associated sequence of multiples wouldn't overlap, and therefore the density of all the numbers was the same.
The 19th-century theorem by the mathematician Mertens allowed Lichtman and Pomerance to take a look at the densities of the primitive set. The maximal value for what the Erds sum of a primitive set could be was given by a special constant. The combined density was the most important factor in determining the Erds sum of a primitive set.
James Maynard, a mathematician at Oxford, said that it was a variation of the original ideas of Erds.
It seemed like the best mathematicians could do that for a long time. How to get that maximum down to 1.64 was not clear. In the meantime, Lichtman graduated and moved to Oxford to do his doctorate, where he worked on other problems related to the primes.
Maynard said it was a complete shock when he suddenly came up with a complete proof.
It was easy to show that the constant 1.78 could be driven down to well below 1.64.
The numbers with prime factors that are close to the primes were different. Lichtman was able to associate several sequence of multiples to each number. The density of all those sequence was the same as before. Some of the space will be taken over by the other multiples.
You can take the number 622. All multiples of 618 have the smallest prime factor. Some of the smaller prime factors could be used to construct the sequence. A sequence may include all the original multiples, but also allow multiples of 618 that are divisible by five. Smaller prime factors are able to be used.
The density of the original multiples was less than 1 because of these additional multiples. It was possible to put a more precise bound on the density.
The worst-case scenario for a primitive set would be between numbers with large prime factors and smaller prime factors. By patching together the two parts of his proof, he was able to show that the Erd's sum is less than 1.64.
There is a numerical moment of truth. I don't know if it's luck or something.
He posted his proof on the internet. The work is striking because it relies on elementary arguments. Thompson said that it wasn't like he was waiting for all the machines to develop. He had some great ideas.
The primes are now considered to be exceptional among the primitive sets, thanks to those ideas. The primes are special. The addition of this adds to their attractiveness.
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