case b = 10, p = 17 gives the cyclic number 0588235294117647; thus 17 is a full reptend prime. Furthermore, 1 divided by 17 written out in base 10 is The case b = 10, p = 19 gives the cyclic number 052631578947368421; thus 19 is a full reptend prime. Furthermore, 1 divided by 19 written out in base 10 is Not all values of p will yield a cyclic number using this formula; for example, p = 13 gives These failed cases will always contain a repetition of digits (possibly several) over the course of p − 1 digits. p = 31 gives These failed cases will always contain a repetition of digits (possibly several) over the course of p − 1 digits. The known pattern to this sequence comes from algebraic number theory, specifically, this sequence is the set of primes p such that 10 is a primitive root modulo p. Artin's conjecture on primitive roots is that this sequence contains 37.395...% of the primes. Patterns of occurrence of full reptend primes Advanced modular arithmetic can show that any prime of the following forms: 40k + 1 40k + 3 40k + 9 40k + 13 40k + 27 40k + 31 40k + 37 40k + 39 can never be a full reptend prime in base 10. The first primes of these forms, with their periods, are: However, studies show that two-thirds of primes of the form 40k + n, where n ∈ {7, 11, 17, 19, 21, 23, 29, 33} are full reptend primes. For some sequences, the preponderance of full reptend primes is much greater. For instance, 285 of the 295 primes of form 120k + 23 below 100000 are full reptend primes, with 20903 being the first that is not full reptend. Binary full reptend primes In base 2, the full reptend primes are: (less than 1000) 3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797, 821, 827, 829, 853, 859, 877, 883, 907, 941, 947, ... For these primes, 2 is a primitive root modulo p, so 2n modulo p can be any natural number