Veit, if at least one side of the inequality takes integer values, say the left side, then you can sometimes convert it to a sequence of integers by replacing f(x) <= g(x) by f(n) <= floor(g(n)), and then look up the nonnegative integer sequence floor(g(n))-f(n) in the OEIS. I once went through Mitrinovic's Handbook of Number Theory and produced a lot of sequences that way - so if your inequality were one of them you would get pointed to a reference. Unfortunately there aren't enough of such sequences in the OEIS - I would like to get more. https://oeis.org/A057641 arises in that way from Lagarias's famous inequality which is equivalent to the Riemann Hyp. Show that sequence is >= 0 and win a million dollars. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Apr 7, 2016 at 3:25 PM, Veit Elser <ve10@cornell.edu> wrote:
How about this for a more manageable, miniature version:
I have an inequality, either one I’ve proven is true or one I suspect is true. How do I discover the name of the inequality, assuming it is true and known, or convince myself that it is a truly new inequality?
Wikipedia has some very long lists, organized in broad categories. Surely there must be something better.
-Veit
On Apr 2, 2016, at 1:35 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
Suppose I had a math problem, and wanted the solution. For instance the following:
How large a subset of the N^2 points on an N-by-N square grid can there be with none of the distances between the points in the subset being equal?
I could try to solve it, or I could ask others, perhaps on this list, if they know the answer.
Thanks to Sloane, there's also a way to look it up: I could "solve" it, by brute force, for the first few N, then look up the resulting sequence in OEIS. If anyone has a general solution, that will point me at it.
Also, if a problem gives rise to a large integer or an unusual real number ("large" and "unusual" meaning it likely never occurs in any other context), I could do a Google search on the integer or a search on one of the several databases of real numbers for the real number.
For instance when I discovered that 8022581057533823761829436662099 was a palindrome in both binary and ternary, a Google search assured me that (probably) nobody had ever noticed this fact before.
Unfortunately, many problems don't lead themselves to any such searches. For instance:
What odd numbers, if any, are equal to the sum of their proper divisors?
If I didn't know that that's called a "perfect number," I'd be unlikely to be able to find a link to the enormous research that's been done on it. (It's very likely the oldest unsolved problem.)
But what's really needed is not just a database of sequences, large integers, or unusual real numbers, but a database of problems. The difficulty, of course, is that there are many different ways of phrasing the same problem, no canonical logical language in which there is a unique way of phrasing a problem (and I'm not sure whether such a language is even possible (Lojban certainly doesn't qualify)) and no known way of automatically translating the former into the latter. But if these difficulties could somehow be solved, it would be a wonderful resource. Each problem would be listed, along with its full or partial solution, whatever is known. If you type in an ambiguous description of the problem, the search program would point out the ambiguity to you.
What do professional mathematicians do when they wonder if a "new" problem has a known solution? I suspect that the division of math into numerous subject areas is of limited utility, given that the same problem may fit in different subject areas, depending on how it's phrased, or (equivalently (?)) may be isomorphic to what sounds like a very different problem.
For instance would anyone have ever noticed the link between the Riemann zeta function and prime numbers if every mathematician knew either number theory or analysis, but never both? I think it was Euler who first noticed the link; he lived early enough that he may have known the whole of then-known math. But that's obviously no longer possible.
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