Positive integers expressible as a sum of three squares in essentially only one way

AbstractThe set S consisting of those positive integers n which are uniquely expressible in the form n = a2 + b2 + c2, a ≧ b ≧ c ≧ 0, is considered. Since n ∈ S if and only if 4n ∈ S, we may restrict attention to those n not divisible by 4. Classical formulas and the theorem that there are only finitely many imaginary quadratic fields with given class number imply that there are only finitely many n ∈ S with n = 0 (mod 4). More specifically, from the existing knowledge of all the imaginary quadratic fields with odd discriminant and class number 1 or 2 it is readily deduced that there are precisely twelve positive integers n such that n ∈ S and n ≡ 3 (mod 8). To determine those n ∈ S such that n ≡ 1, 2, 5, 6 (mod 8) requires the determination of the imaginary quadratic fields with even discriminant and class number 1, 2, or 4. While the latter information is known empirically, it has not been proved that the known list of 33 such fields is complete. If it is complete, then our arguments show that there are exactly 21 positive integers n such that n ∈ S and n ≡ 1, 2, 5, 6 (mod 8)