Slashdot Mirror
Domain: aimath.org
Stories and comments across the archive that link to aimath.org.
Stories · 6
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Euler's Partition Function Theory Finished
universegeek writes "Mathematician Ken Ono, from Emory, has solved a 250-year-old problem: how to exactly and explicitly generate partition numbers. Ono and colleagues were able to finally do this by realizing that the pattern of partition numbers is fractal (PDF). This pattern allowed them to find a finite, algebraic formula, which is like striking oil in mathematics." -
Finding the First Trillion Congruent Numbers
eldavojohn writes "First stated by al-Karaji about a thousand years ago, the congruent number problem is simplified to finding positive whole numbers that are the area of a right triangle with rational number sides. Today, discovering these congruent numbers is limited only by the size of a computer's hard drive. An international team of mathematicians recently decided to push the limits on finding congruent numbers and came up with the first trillion. Their two approaches are outlined in detail, with pseudo-code, in their paper (PDF) as well as details on their hardware. For those of you familiar with this sort of work, the article provides links to solving this problem — from multiplying very large numbers to identifying square-free congruent numbers." -
Finding the First Trillion Congruent Numbers
eldavojohn writes "First stated by al-Karaji about a thousand years ago, the congruent number problem is simplified to finding positive whole numbers that are the area of a right triangle with rational number sides. Today, discovering these congruent numbers is limited only by the size of a computer's hard drive. An international team of mathematicians recently decided to push the limits on finding congruent numbers and came up with the first trillion. Their two approaches are outlined in detail, with pseudo-code, in their paper (PDF) as well as details on their hardware. For those of you familiar with this sort of work, the article provides links to solving this problem — from multiplying very large numbers to identifying square-free congruent numbers." -
Twin Prime Proof Erroneous
mindriot writes "The fairly recent perceived breakthrough in prime number theory regarding twin primes, as mentioned on slashdot, is apparently not quite perfect: 'On April 23rd, Andrew Granville of the Universite de Montreal and K. Soundararajan of the University of Michigan found a technical difficulty buried in one of the arguments in the preprint of Goldston and Yildrim. The main issue is that some quantities which were believed to be small error terms are actually the same order of magnitude as the main term. For now this difficulty remains unresolved.' A more detailed technical description is also available." -
Twin Prime Proof Erroneous
mindriot writes "The fairly recent perceived breakthrough in prime number theory regarding twin primes, as mentioned on slashdot, is apparently not quite perfect: 'On April 23rd, Andrew Granville of the Universite de Montreal and K. Soundararajan of the University of Michigan found a technical difficulty buried in one of the arguments in the preprint of Goldston and Yildrim. The main issue is that some quantities which were believed to be small error terms are actually the same order of magnitude as the main term. For now this difficulty remains unresolved.' A more detailed technical description is also available." -
Another Breakthrough in Prime Number Theory
Battal Boy writes "From aimath.org: Dan Goldston and his Turkish colleague Yalcin Cem Yildirim have smashed all previous records on the size of small gaps between prime numbers. This work is a major step toward the centuries-old problem of showing that there are infinitely many 'twin primes': prime numbers which differ by 2, such as 11 and 13, 17 and 19, 29 and 31,...I am especially proud of this achievement as Yalcin is a close friend of mine from way back! You may also want to check out the Mercury News Article and Dan Goldston's home page where you can see a photo of Dan's back being slowly but surely broken by two of his children ..." Finding patterns in primes seems to be all the rage.