Great idea! The current system also works fairly well: First google and, if you have access, use MathSciNet to search for papers related to your keywords, with which you can experiment. If no luck, ask a mathematician who's in the general field your problem falls in if they know whether it's been solved, and if not, do they know someone else more likely to know whether it's been solved. HOWEVER, if the problem falls into two or more disparate fields, then few or none might know if it's been solved. And, if the problem is sufficiently interesting but unsolved, there's a fair chance someone is currently working on it and may not be willing to discuss their progress until such time as it's published. OK, suppose a database of published results is published, like MathSciNet but better. How could it be organized better than MathSciNet? Unfortunately, that database is very stingy about who gets access to it. —Dan

On Apr 2, 2016, at 10:35 AM, 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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You might find "Fingerprint Databases for Theorems" by Billey and Tenner ( https://www.math.washington.edu/~billey/papers/fingerprints.pdf) of interest. Jim Propp On Saturday, April 2, 2016, 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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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.