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 rudy235 2022-06-07 22:55

Graduate Student’s Side Project Proves Prime Number Conjecture

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Disclosure: i must say do not like the heading too much but that is the actual title of the Article in Quanta magazine with yesterday’s date by Jordana Cepelewicz.

[SIZE="4"]Graduate Student’s Side Project Proves Prime Number Conjecture

[SIZE="3"][I]Jared Duker Lichtman, 26, has proved a longstanding conjecture relating prime numbers to a broad class of “primitive” sets. To his adviser, it came as a “complete shock.”[/I][/SIZE][/SIZE]

[SIZE="2"]Primitive sets are sequences of numbers in which no number can divide any other number. In this universe of sets, the primes are unique.[/SIZE]

[B]Jordana Cepelewicz
Senior Writer[/B]

June 6, 2022

As the atoms of arithmetic, prime numbers have always occupied a special place on the number line. Now, Jared Duker Lichtman, a 26-year-old graduate student at the University of Oxford, has resolved a well-known conjecture, establishing another facet of what makes the primes special — and, in some sense, even optimal. “It gives you a larger context to see in what ways the primes are unique, and in what ways they relate to the larger universe of sets of numbers,” he said.

The conjecture deals with primitive sets — sequences in which no number divides any other. Since each prime number can only be divided by 1 and itself, the set of all prime numbers is one example of a primitive set. So is the set of all numbers that have exactly two or three or 100 prime factors.

Primitive sets were introduced by the mathematician Paul Erdős in the 1930s. At the time, they were simply a tool that made it easier for him to prove something about a certain class of numbers (called perfect numbers) with roots in ancient Greece. But they quickly became objects of interest in their own right — ones that Erdős would return to time and again throughout his career.

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 Dr Sardonicus 2022-06-07 23:49

[QUOTE=rudy235;607319]Disclosure: i must say do not like the heading too much but that is the actual title of the Article in Quanta magazine with yesterday’s date by Jordana Cepelewicz.
<snip>[/QUOTE]I agree, [i]Quanta[/i] booted the title. A more accurate title appears to be [url=https://arxiv.org/abs/2202.02384]A proof of the Erdős primitive set conjecture[/url] (Arxiv preprint).

I found a description of the conjecture online:[quote]A subset of the integers larger than 1 is primitive if no member divides another. Erdős proved in 1935 that $$\sum_{a\in A}\frac{1}{a\log(a)}$$ for [i]a[/i] running over a primitive set [i]A[/i] is universally bounded over all choices for [i]A[/i]. In 1988 he asked if this universal bound is attained for the set of prime numbers.[/quote]

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