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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."
I'm so not a maths geek, but why is this useful other than being able to say "hey, we found the first trillion congruent numbers"?
I know that certain branches are useful for cryptography purposes, but what awesomeness does this let us do?
It's all fun and games until a 200' robot dinosaur shows up and trashes Neo-Tokyo... Again
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.
They aren't hiding any part of their methodology. They've given more than enough details. Posting the actual code in the paper would be a distraction. When publishing new algorithms mathematicians generally outline it in pseudocode since this is a) easier to follow and b) much more useful for issues of proving formal correctness. I would not be at all surprised if you can find the code on the website of one of the authors and almost certainly the authors will provide the code details if you send them an email.
I don't believe they found the first trillion congruent numbers; rather, they tested the first trillion whole numbers for congruency.
The rest was already full of pr0n. These guys don't get out much.