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@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).

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.

-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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