OK, here is a proof of the 2nd conjecture that I claimed before (but I should have said "positive definite" quadratic forms) THEOREM: Let Q be a positive-definite integer-valued quadratic form in some number of integer variables. Then either Q fails to represent an infinite subset of natural numbers, or Q represents each member of an (other) infinite subset of them in an unboundedly large number of ways, indeed growing like a power law. PROOF (OF THIS PLUS SOME MORE): Via the Hardy-Littlewood "singular series" one can prove (i.e. others have proven) that the number of representations of N as a sum of s terms, each a kth power of a natural number, behaves for N large like PositiveConstant(s,k)*N^(s/k +- o(1)) provided s is large enough compared to k. (And precise criteria are available for how large s must be.) But it is simpler if we do not ask for #representations = something+-small error, but instead merely ask that #reps >= PositiveConstant(s,k)*N^(s/k) and merely for an infinite subset of (but not all) N>0. Then this is trivially true by simply counting the number of ways you can choose s integers in the set [1/2, 1]*N^(1/k). There are order N^(s/k) ways, and they all produce reps of numbers in the set [1/2^k, 1]*s*N. Hence by pigeonhole principle some number in this set must be represented at least (1/s)*N^(s/k - 1) times. This count goes to infinity (and like a power law) when N-->infinity, provided only that s>k, In particular, any ternary (or quaternary,... or s-variate for any s>=3) integral positive definite quadratic form represents at least some integers (forming an infinite subset of natural numbers) in unboundedly many ways; the proof is similar by now counting lattice points in a ball of radius squareroot(N), and using pigeonhole principle. (Using the s-dimensional lattice corresponding to whatever s-variate quadratic form we are speaking of.) And "unboundedly many" is, more precisely, a power law. Now, what about quadratic forms which are not necessarily positive definite, or might be binary (2-variable) or only unary (1-variable)? Clearly a unary quadratic form Q(x)=a*x^2 cannot represent all integers, albeit those it does represent, are only represented in 2 ways. The binary *indefinite* form Q(x,y)=x*y represents each nonzero integer N in 4*#divisors(N) ways, which is always >=4 (and equals 4 if N=prime), and which is unboundedly large for highly composite N, but note the #divisors(N) function is upper bounded by N^o(1), i.e. grows slower than any positive power of N. [Jeff D. Caldwell also was suggested to me, erroneously, that you could say the same about the form x^2-y^2 = (x-y)*(x+y). That does not quite work because x-y and x+y necessarily have the same parity, hence this form cannot represent integers congruent to 2 mod 4. But his idea does work if applied to x*y.] A binary form Q(x,y)=a*x*x+c*y*y+b*x*y is positive definite if b*b < 4*a*c. I claim that every positive definite binary integral quadratic form (a) necessarily leaves an infinite subset of natural numbers unrepresented, and (b) necessarily represents some integers an unboundedly large number of ways. To prove (a), which was already known to C.F.Gauss and P.Fermat, let the "discriminant" of the form be d=4*a*c-b*b<0 and note that integers N cannot be represented if d is nonsquare modulo 4N. By quadratic reciprocity, this is an infinite positive density subset of N. Also (a) apparently was addressed in this paper: Valentin Blomer, Andrew Granville: Estimates for representation numbers of quadratic forms, Duke Math'l. J. 135,2 (2006) 261-302. which I think claims to prove that for any positive definite integer-valued binary quadratic form, the number of represented naturals below N is at least asymptotic to PositiveConstant * N / log(N)^(1/2). [But I have not actually read this paper -- can anybody provide PDF?] To prove (b) --which I provide as a bonus, since I do not actually need it -- consider the lattice of the complex numbers whose squared-norms are the quadratic form... this is a number ring... and my point is that that you can multiply complex numbers in the ring to get a norm which is represented in many ways. For example as the simplest case so you can see what I am talking about, Q(x,y) = x*x+y*y, the lattice is the just the Gaussian integers z=a+i*b with a,b ordinary integers, and then Q(x,y)=|z|^2. Now any norm is represented in 8 ways via a+ib, a-ib, b+ia, b-ia, -a-ib, -a+ib, -b+ia, -b-ia (well, only 4 if a=b, but ignore that as non-generic) but if we multiply (a+ib) * (c+id) then we get a norm generically represented in 16 ways, and (a+ib)*(c+id)*(e+if) will have norm generically represented in 32 ways, etc. My point is, in this manner you can construct numbers which have unboundedly many representations as sums of two squares; they arise from highly-composite Gaussian integers. Now similarly, if we work over any other complex number ring in place of "the Gaussian integers" we can do the same kind of construction -- highly composite ring members will yield natural numbers with unboundedly many representations by whatever positive definite binary quadratic form corresponds to the norm in that ring. This proves (b), indeed this construction should prove the maximum number (among natural numbers n<N) of representations of n must grow at least as quickly as 2^([1/2 - o(1)] * lnN / lnlnN) when N-->infinity. Q.E.D. EXTENSION: Theorem also valid for rational-valued quadratic forms with integer arguments. (Because just multiply by some constant denominator to get an integer valued form...) -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)