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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."
Among other issues, which numbers are congruent numbers is deeply related to the Birch-Swinnerton-Dyer conjecture which is a major open problem http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture. This is due to a theorem which relates whether a number is a congruent number or not to the number of solutions of certain ternary quadratic forms.
The summary isn't quite accurate in that regard: The problem of finding congruent numbers is not completely solved. If BSD is proven then we can reasonably call the question solved. But it doesn't look like there's much hope for anyone resolving BSD in the foreseeable future. There's also hope that the data will give us further patterns and understanding of ternary quadratic forms and related issues which show up in quite a few natural contexts (such as Julia Robinson's proof that first order Q is undecidable).
FtFP (From the F***ing Paper):
We report on a computation of congruent numbers, which subject to the Birch and Swinnerton-Dyer conjecture is an accurate list up to 10^12.