[math-fun] some conjectures (theorems?) about quadratic forms
Jeff D Caldwell: How about ax^2 + bxy + cy^2 with a = -1, b = 1 and c = 0 for conjecture #2? The form generates every natural exactly the number of times as the number of factors of that natural, counting such that a prime squared has 3 factors. (The form also generates integers less than 0.) On Fri, Jun 27, 2014 at 11:03 AM, Warren D Smith <warren.wds at gmail.com> wrote:
D.Lehmer showed that there is an infinite set of numbers representable as a sum of 4 squares in exactly 1 way (up to re-ordering), http://oeis.org/A006431
1. I conjecture there does NOT exist any integer ternary quadratic form Q, for which there is an infinite set of numbers representable as Q(a,b,c) but only in a bounded nonzero number of ways. In fact, if N is representable N=Q(a,b,c), then I conjecture N is representable in at least N^0.499 ways (for all sufficiently large N).
--WDS: JDC's example looks good, except it is a binary quadratic form Q(x,y), and I wanted ternary Q(x,y,z).
On Fri, Jun 27, 2014 at 5:18 PM, Warren D Smith <warren.wds@gmail.com> wrote:
Jeff D Caldwell: How about ax^2 + bxy + cy^2 with a = -1, b = 1 and c = 0 for conjecture #2? The form generates every natural exactly the number of times as the number of factors of that natural, counting such that a prime squared has 3 factors. (The form also generates integers less than 0.)
On Fri, Jun 27, 2014 at 11:03 AM, Warren D Smith <warren.wds at gmail.com> wrote:
D.Lehmer showed that there is an infinite set of numbers representable as a sum of 4 squares in exactly 1 way (up to re-ordering), http://oeis.org/A006431
1. I conjecture there does NOT exist any integer ternary quadratic form Q, for which there is an infinite set of numbers representable as Q(a,b,c) but only in a bounded nonzero number of ways. In fact, if N is representable N=Q(a,b,c), then I conjecture N is representable in at least N^0.499 ways (for all sufficiently large N).
--WDS: JDC's example looks good, except it is a binary quadratic form Q(x,y), and I wanted ternary Q(x,y,z).
As my first sentence said, I was responding to your conjecture #2. You quoted your conjecture #1 in your response to me. Conjecture #1 does indeed ask for ternary quadratic forms. For convenience, I quote your conjecture #2 in full: "2. I conjecture (actually, for this one I have a proof sketch, but I have not checked its details carefully) that there does not exist any integer quadratic form, in any finite number of variables, for which both (a) every natural number is representable (b) but only in a bounded number of ways. In fact I think my proof will show that if (a), then the number of representations must, for an infinite subset of N, grow faster than some positive power of N." I took binary quadratic forms to be a quadratic form in a finite number of variables, and my form does (a) in that every natural number is represented, and (b) every natural is represented in a bounded number of ways. Perhaps I misunderstood something? That often happens with me. I also said my form produces each natural exactly the number of times as that natural has factors. However, allowing for negative values of x, the naturals are duplicated or mirrored, according to their appearances with positive values of x. So each natural appears twice as often as the number of factors of the natural. Jeff
participants (2)
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Jeff Caldwell -
Warren D Smith