The modular ring Z[subscript 6] defines integers via ( 6r[subscript i] + ( i - 3)) where i is the class and r[subscript i] the row when tabulated in an array. Since only Classes 2[subscipt 6] and 4[subscript 6] contain odd primes, this modular ring is ideally suited to the analysis of twin primes. The calculations are facilitated by the use of the right-end digit (RED) technique. (Contains 3 tables.)