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).