Rings as Groups with Operators

(11) KM s 0 (mod 2*--9). Conversely (11) implies (9). Since (9) holds for the modulus 2-9ikf, it follows similarly that (11) holds for the modulus 2~-9 with ikf = 2ikfi. Hence (11) will be true for the given modulus if M = 2~M\. This supplies a proof by induction that (8) is a universal form for every w ^ 4 . If, in addition,* ikf is divisible by every prime p where 3<p^n, we satisfy the necessary condition given by Dickson f for the form (8) to represent at least one set of n primes. The proof of the sufficiency of this condition still remains a challenge to the ingenuity of number theorists.