Full reptend prime

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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.

Base 2 full reptend primes[edit]

In base 2 the full reptend primes are:

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, ... (sequence A001122 in OEIS)

For this primes, 2 is a primitive root modulo p.

All of them are of form 8k+3 or 8k+5, because if p = 8k+1 or 8k+7, then 2 is a quadratic residue modulo p, so p divides 2(p-1)/2−1, and the period of 1/p in base 2 must divide (p−1)/2 and cannot be p−1, so they are not long primes.

8k+1 17 41 73 89 97 113 137 193 233 241 257 281 313 337 353 401 409 433 449 457 521 569 577 593 601 617 641 673
period 8 20 9 11 48 28 68 96 29 24 16 70 156 21 88 200 204 72 224 76 520 284 144 148 25 154 64 48
8k+7 7 23 31 47 71 79 103 127 151 167 191 199 223 239 263 271 311 359 367 383 431 439 463 479 503 599 607 631
period 3 11 5 23 35 39 51 7 15 83 95 99 37 119 131 135 155 179 183 191 43 73 231 239 251 299 303 45

References[edit]