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Old 2018-07-30, 18:41   #15
CRGreathouse
 
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Aug 2006

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Quote:
Originally Posted by jwaltos View Post
Regarding the questions of the "general solution of the integer factorization problem" and the "distribution of the prime numbers", these are tough questions and I was wondering if there was any noteworthy conceptual breakthrough of any kind regarding these questions.
There have been tremendous breakthroughs, conceptual and otherwise, on both problems.

For an overview of the first see
https://www.ams.org/notices/199612/pomerance.pdf

After that I think
https://maths-people.anu.edu.au/~bre...phy-thesis.pdf
is standard, perhaps with something like
http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf
as a gentler transition. Once you've absorbed Murphy's thesis there are several people here who can advise on the best papers to read to better understand state-of-the-art integer factorization.

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As for the distribution of prime numbers, that's a much broader subject. The first major result was the Prime Number Theorem at the close of the 19th century. A recent breakthrough was Zhang's bounded gap theorem. I wrote a post diagramming its relationship to other results; you might find it useful.
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