Volume 11, 2005, Number 2

Modular rings are used to analyse integers of the form N = 2^m + 1. When m is odd, the integer structure prevents the formation of primes. When m is even, N ‘commonly’ has a right-end-digit of 5 and so is not a prime then. However, a sequence defined by m = 4 + 4q, q = 0, 1, 2, 3 can generate some primes as the right-end-digit is 7. Elements of this sequence satisfy the non-linear recurrence relation G_m = G_m−1² − 2G_m−1 + 2. Fermat numbers, where m = 2ⁿ satisfy this recurrence relation. However, in this case, the integer structure reveals that primes are limited to n < 5.