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- C. Caldwell, "Unique (period) primes and the factorization of cyclotomic polynomial minus one," Mathematica Japonica, 46:1 (1997) 189--195. MR 99b:11139
Let b be an integer greater than one. For each prime p not dividing b, the expansion of 1/p (radix b) has a period dividing p-1. Unique primes are those primes (not dividing b) for which no other prime has the same period. Let Φn(x) be the nth cyclotomic polynomial. We develop criteria for when Φn(x)-1 is divisible by Φm(x), and use this to add 18 new primes (and 10 new probable primes) to the list of the 29 previously known unique primes (base 10). The divisibility properties found are useful in the search for primes of many forms.