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Theorem If b > 1 and bN + 1 is prime, then b is even and N = 2n. For if b is odd then bN + 1 is even; and if N has an odd factor k and N = kl, then bN + 1 is divisible by bl + 1. All the primes of the form bN + 1 (b > 1) are some Fermat (b = 2) or Generalized Fermat primes. Notation: GF(n, b) = bN + 1, with N = 2n. Remark: The polynomial xN + 1 is the 2(n+1)th cyclotomic polynomial, thus xN + 1 is irreducible. |