Apparently he intended to ask a question equivalent to the following: Is there a way to express every integer N>=0 in the form a^2 + b^2 - c^2 where a,b,c integers and c^2 <= a^2+b^2 < (c+1)^2 ? An integer S is expressible as a^2+b^2 if and only if its prime divisors of the form 4m+3 have even powers inside S's prime factorization. In particular, every number whose prime factors are 4m+1 primes and the prime 2, only, is thus-expressible. So for example Wilson's question would have a "yes" answer if CONJECTURE: every number N>0 is expressible as 2^k * P - c^2 where P is a prime of form 4m+1, k is in {0,1,2}, and c is integer, and c^2 < 2^k * P < (c+1)^2. Well, what can I say. This conjecture seems extremely likely to be true, in fact I would think every N>0 is thus-expressible in an infinite number of ways. If there were any N only thus-expressible in a finite number of ways, that would be an astonishing find that would win you a Fields medal immediately. But I do not see a way to prove it. Perhaps of some relevance: The ternary form a^2 + b^2 - c^2 is known to represent all integers [Leonard Eugene Dickson: Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939, theorem 109? A.Oppenheim: The determination of all universal ternary quadratic forms, Quart. J. Math. 1 (1930) 179-185] -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)