Full reptend prime

In number theory, a full reptend prime or long prime in base b is a prime number p such that the formula

\frac{b^{p - 1} - 1}{p}

(where p does not divide b) gives a cyclic number. Therefore the digital expansion of 1/p in base b repeats the digits of the corresponding cyclic number infinitely. Base 10 may be assumed if no base is specified.

The first few values of p for which this formula produces cyclic numbers in decimal are (sequence A001913 in OEIS)

7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983 …

For example, the case b = 10, p = 7 gives the cyclic number 142857, thus, 7 is a full reptend prime. Furthermore, 1 divided by 7 written out in base 10 is 0.142857142857142857142857...

Not all values of p will yield a cyclic number using this formula; for example p = 13 gives 076923076923. These failed cases will always contain a repetition of digits (possibly several).

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.

The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers."

The corresponding cyclic number to prime p will possess p - 1 digits if and only if p is a full reptend prime.

Patterns of occurrence of full reptend primes

Advanced modular arithmetic can show that any prime of the following forms:

  1. 40k+1
  2. 40k+3
  3. 40k+9
  4. 40k+13
  5. 40k+27
  6. 40k+31
  7. 40k+37
  8. 40k+39

can never be a full reptend prime in base-10. The first primes of these forms, with their periods, are:

40k+1 40k+3 40k+9 40k+13 40k+27 40k+31 40k+37 40k+39
41
period 5		 43
period 21		 89
period 44		 13
period 6		 67
period 33		 31
period 15		 37
period 3		 79
period 13
241
period 30		 83
period 41		 409
period 204		 53
period 13		 107
period 53		 71
period 35		 157
period 78		 199
period 99
281
period 28		 163
period 81		 449
period 32		 173
period 43		 227
period 113		 151
period 75		 197
period 98		 239
period 7
401
period 200		 283
period 141		 569
period 284		 293
period 146		 307
period 153		 191
period 95		 277
period 69		 359
period 179

However, studies show that two-thirds of primes of the form 40k+n, where n ≠ {1,3,9,13,27,31,37,39} 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.

References