Legendre-style ternary construction blocked 2-adically: let p be even

When proving a ternary form represents m by building [[a,b,c],[b,p,0],[c,0,m]] with a*p = b^2 + d and det = d*m - c^2*p, the usual recipe takes p an odd Dirichlet prime and uses it as the odd value of the form. For some residue classes (here m = 14 mod 16 in determinant 10) no such form exists for any c, which a quick brute-force search confirms; the obstruction is 2-adic. Fix: take p = 2q with q prime and d = 2 mod 4. Make the root b of -d mod q even (replace b by b + q), so 2q | b^2 + d and a = (b^2+d)/(2q) = (2t^2 + d/2)/q is odd automatically. a then plays the role of the odd value. Also, if m is divisible by the prime in the determinant, the det equation forces that prime to divide c, and the condition 'd is a non-residue' then holds exactly for the non-exceptional m.