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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
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.
That's a good explanation. I have to emphasize though, that they actually found all the congruent numbers up to a trillion only under the completely unproven hypothesis that the Birch and Swinnerton-Dyer conjecture is true. It's entirely possible that this conjecture is false, and some of the numbers they found are actually not congruent numbers. However, part of the conjecture is known (by work of Coates and Wiles -- the same Wiles who proved Fermat's Last Theorem), so we do know that all numbers they didn't list are definitely not congruent numbers.