In number theory, a full reptend prime, full repetend prime, proper prime[1] :166 or long prime in base b is a prime number p such that the formula
(where p does not divide b) gives a cyclic number. Therefore the digital expansion of in base b repeats the digits of the corresponding cyclic number infinitely, as does that of
with rotation of the digits for any a between 1 and p - 1. The cyclic number corresponding to prime p will possess p - 1 digits if and only if p is a full reptend prime. That is, ordbp = p - 1.
Base 10 may be assumed if no base is specified, in which case the expansion of the number is called a repeating decimal. In base 10, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., 9 appears in the repetend the same number of times as each other digit.[1]:166
The values of p less than 1000 for which this formula produces cyclic numbers in decimal are:
- 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, ... (sequence A001913 in OEIS)
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 0.142857 142857 142857 142857...
Not all values of p will yield a cyclic number using this formula; for example p = 13 gives 076923 076923. 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.
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."
Contents
Patterns of occurrence of full reptend primes[edit]
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:
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: (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, ... (sequence A001122 in OEIS)
For these primes, 2 is a primitive root modulo p, so 2n module p can be any natural number between 1 and p-1.
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 full reptend primes in base 2.
Further, all safe primes congruent to 3 (mod 8) are full reptend primes in base 2. For example, 3, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, etc. (less than 1500)
The following is a list about the periods to the primes congruent to 1 or 7 (mod 8): (less than 1000)
8k + 1 | 17 | 41 | 73 | 89 | 97 | 113 | 137 | 193 | 233 | 241 | 257 | 281 | 313 | 337 | 353 | 401 | 409 | 433 | 449 | 457 | 521 | 569 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
period | 8 | 20 | 9 | 11 | 48 | 28 | 68 | 96 | 29 | 24 | 16 | 70 | 156 | 21 | 88 | 200 | 204 | 72 | 224 | 76 | 260 | 284 |
8k + 1 | 577 | 593 | 601 | 617 | 641 | 673 | 761 | 769 | 809 | 857 | 881 | 929 | 937 | 953 | 977 | 1009 | 1033 | 1049 | 1097 | 1129 | 1153 | 1193 |
period | 144 | 148 | 25 | 154 | 64 | 48 | 380 | 384 | 404 | 428 | 55 | 464 | 117 | 68 | 488 | 504 | 258 | 262 | 274 | 564 | 288 | 298 |
8k + 7 | 7 | 23 | 31 | 47 | 71 | 79 | 103 | 127 | 151 | 167 | 191 | 199 | 223 | 239 | 263 | 271 | 311 | 359 | 367 | 383 | 431 | 439 |
period | 3 | 11 | 5 | 23 | 35 | 39 | 51 | 7 | 15 | 83 | 95 | 99 | 37 | 119 | 131 | 135 | 155 | 179 | 183 | 191 | 43 | 73 |
8k + 7 | 463 | 479 | 487 | 503 | 599 | 607 | 631 | 647 | 719 | 727 | 743 | 751 | 823 | 839 | 863 | 887 | 911 | 919 | 967 | 983 | 991 | 1031 |
period | 231 | 239 | 243 | 251 | 299 | 303 | 45 | 323 | 359 | 121 | 371 | 375 | 411 | 419 | 431 | 443 | 91 | 153 | 483 | 491 | 495 | 515 |
The binary period of nth prime are
- 2, 4, 3, 10, 12, 8, 18, 11, 28, 5, 36, 20, 14, 23, 52, 58, 60, 66, 35, 9, 39, 82, 11, 48, 100, 51, 106, 36, 28, 7, 130, 68, 138, 148, 15, 52, 162, 83, 172, 178, 180, 95, 96, 196, 99, 210, 37, 226, 76, 29, 119, 24, 50, 16, 131, 268, 135, 92, 70, 94, 292, 102, 155, 156, 316, 30, 21, 346, 348, 88, 179, 183, 372, 378, 191, 388, 44, ... (this sequence starts at n = 2, or the prime = 3) (sequence A014664 in OEIS)
The binary period level of nth prime are
- 1, 1, 2, 1, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 2, 8, 2, 1, 8, 2, 1, 2, 1, 3, 4, 18, 1, 2, 1, 1, 10, 3, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 6, 1, 3, 8, 2, 10, 5, 16, 2, 1, 2, 3, 4, 3, 1, 3, 2, 2, 1, 11, 16, 1, 1, 4, 2, 2, 1, 1, 2, 1, 9, 2, 2, 1, 1, 10, 6, 6, 1, 2, 6, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, ... (sequence A001917 in OEIS)
However, studies show that three-fourths of primes of the form 8k+n, where n ≠ {1,7} are full reptend primes in base 2 (For example, there are 87 primes below 1000 congruent to 3 or 5 (mod 8), and 67 of them are full-reptend in base 2, it is total 77%). For some sequences, the preponderance of full reptend primes is much greater. For instance, all of the primes of form 24k+5 below 1000 are full reptend primes in base 2, with 1013 being the first that is not full reptend in base 2.
Full reptend primes in various bases[edit]
Artin also conjectured that full-reptend primes in all bases except perfect squares and −1 comprise 37.395...% of all primes.
Base | Full reptend primes | OEIS sequence |
---|---|---|
−20 | 11, 13, 17, 31, 37, 53, 59, 73, 79, 113, 131, 137, 139, 157, 173, 179, 191, 199, 211, 233, 239, 257, 271, 277, ... | A105892 |
−18 | 5, 7, 23, 29, 31, 37, 47, 53, 61, 71, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 223, 239, ... | A105890 |
−16 | 3, 7, 11, 19, 23, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 163, 167, 179, 191, 199, 211, 227, 239, 263, 271, ... | A105876 |
−12 | 5, 17, 23, 41, 47, 53, 59, 71, 83, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 239, 251, 257, ... | A105884 |
−11 | 2, 7, 13, 17, 29, 41, 73, 79, 83, 101, 107, 109, 127, 131, 139, 149, 151, 167, 173, 197, 227, 233, 239, 263, ... | A105883 |
−10 | 3, 17, 29, 31, 43, 61, 67, 71, 83, 97, 107, 109, 113, 149, 151, 163, 181, 191, 193, 199, 227, 229, 233, 257, ... | A007348 |
−9 | 2, 7, 11, 19, 23, 31, 43, 47, 59, 71, 79, 83, 107, 127, 131, 139, 163, 167, 179, 191, 199, 211, 223, 227, 239, ... | A105881 |
−8 | 5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 461, 479, 503, ... | A105880 |
−7 | 2, 3, 5, 13, 17, 31, 41, 47, 59, 61, 83, 89, 97, 101, 103, 131, 139, 167, 173, 199, 227, 229, 241, 251, 257, ... | A105879 |
−6 | 13, 17, 19, 23, 41, 47, 61, 67, 71, 89, 109, 113, 137, 157, 167, 211, 229, 233, 257, 263, 277, 283, 331, 359, ... | A105878 |
−5 | 2, 11, 17, 19, 37, 53, 59, 73, 79, 97, 113, 131, 137, 139, 151, 157, 173, 179, 193, 197, 233, 239, 257, 277, ... | A105877 |
−4 | 3, 7, 11, 19, 23, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 163, 167, 179, 191, 199, 211, 227, 239, 263, 271, ... | A105876 |
−3 | 2, 5, 11, 17, 23, 29, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, ... | A105875 |
−2 | 5, 7, 13, 23, 29, 37, 47, 53, 61, 71, 79, 101, 103, 149, 167, 173, 181, 191, 197, 199, 239, 263, 269, 271, 293, ... | A105874 |
−1 | 2, 3 | |
0 | None | |
1 | 2 | |
2 | 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, ... | A001122 |
3 | 2, 5, 7, 17, 19, 29, 31, 43, 53, 79, 89, 101, 113, 127, 137, 139, 149, 163, 173, 197, 199, 211, 223, 233, 257, ... | A019334 |
4 | None | |
5 | 2, 3, 7, 17, 23, 37, 43, 47, 53, 73, 83, 97, 103, 107, 113, 137, 157, 167, 173, 193, 197, 223, 227, 233, 257, ... | A019335 |
6 | 11, 13, 17, 41, 59, 61, 79, 83, 89, 103, 107, 109, 113, 127, 131, 137, 151, 157, 179, 199, 223, 227, 229, 233, ... | A019336 |
7 | 2, 5, 11, 13, 17, 23, 41, 61, 67, 71, 79, 89, 97, 101, 107, 127, 151, 163, 173, 179, 211, 229, 239, 241, 257, ... | A019337 |
8 | 3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 269, 293, 317, 347, 389, 419, 443, 461, 467, ... | A019338 |
9 | 2 | |
10 | 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, ... | A001913 |
11 | 2, 3, 13, 17, 23, 29, 31, 41, 47, 59, 67, 71, 73, 101, 103, 109, 149, 163, 173, 179, 197, 223, 233, 251, 277, ... | A019339 |
12 | 5, 7, 17, 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 223, 257, 269, 281, 283, ... | A019340 |
13 | 2, 5, 11, 19, 31, 37, 41, 47, 59, 67, 71, 73, 83, 89, 97, 109, 137, 149, 151, 167, 197, 227, 239, 241, 281, 293, ... | A019341 |
14 | 3, 17, 19, 23, 29, 53, 59, 73, 83, 89, 97, 109, 127, 131, 149, 151, 227, 239, 241, 251, 257, 263, 277, 283, 307, ... | A019342 |
15 | 2, 13, 19, 23, 29, 37, 41, 47, 73, 83, 89, 97, 101, 107, 139, 149, 151, 157, 167, 193, 199, 227, 263, 269, 271, ... | A019343 |
16 | None | |
17 | 2, 3, 5, 7, 11, 23, 31, 37, 41, 61, 97, 107, 113, 131, 139, 167, 173, 193, 197, 211, 227, 233, 269, 277, 283, ... | A019344 |
18 | 5, 11, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 139, 149, 157, 163, 173, 179, 181, 197, 227, 251, 269, ... | A019345 |
19 | 2, 7, 11, 13, 23, 29, 37, 41, 43, 47, 53, 83, 89, 113, 139, 163, 173, 191, 193, 239, 251, 257, 263, 269, 281, ... | A019346 |
20 | 3, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 103, 107, 113, 137, 157, 163, 167, 173, 223, 227, 233, 257, 263, 277, ... | A019347 |
21 | 2, 19, 23, 29, 31, 53, 71, 97, 103, 107, 113, 137, 139, 149, 157, 179, 181, 191, 197, 223, 233, 239, 263, 271, ... | A019348 |
22 | 5, 17, 19, 31, 37, 41, 47, 53, 71, 83, 107, 131, 139, 191, 193, 199, 211, 223, 227, 233, 269, 281, 283, 307, ... | A019349 |
23 | 2, 3, 5, 17, 47, 59, 89, 97, 113, 127, 131, 137, 149, 167, 179, 181, 223, 229, 281, 293, 307, 311, 337, 347, ... | A019350 |
24 | 7, 11, 13, 17, 31, 37, 41, 59, 83, 89, 107, 109, 113, 137, 157, 179, 181, 223, 227, 229, 233, 251, 257, 277, ... | A019351 |
25 | 2 | |
26 | 3, 7, 29, 41, 43, 47, 53, 61, 73, 89, 97, 101, 107, 131, 137, 139, 157, 167, 173, 179, 193, 239, 251, 269, 271, ... | A019352 |
27 | 2, 5, 17, 29, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, ... | A019353 |
28 | 5, 11, 13, 17, 23, 41, 43, 67, 71, 73, 79, 89, 101, 107, 173, 179, 181, 191, 229, 257, 263, 269, 293, 313, 331, ... | A019354 |
29 | 2, 3, 11, 17, 19, 41, 43, 47, 73, 79, 89, 97, 101, 113, 127, 131, 137, 163, 191, 211, 229, 251, 263, 269, 293, ... | A019355 |
30 | 11, 23, 41, 43, 47, 59, 61, 79, 89, 109, 131, 151, 167, 173, 179, 193, 197, 199, 251, 263, 281, 293, 307, 317, ... | A019356 |
32 | 3, 5, 13, 19, 29, 37, 53, 59, 67, 83, 107, 139, 149, 163, 173, 179, 197, 227, 269, 293, 317, 347, 349, 373, 379, ... | A019358 |
36 | None | |
40 | 7, 11, 17, 19, 23, 29, 47, 59, 73, 97, 101, 103, 109, 131, 137, 139, 149, 167, 179, 193, 229, 233, 257, 263, ... | A019365 |
48 | 5, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 127, 137, 149, 151, 173, 197, 199, 211, 223, 233, 257, 269, ... | A019373 |
50 | 3, 11, 13, 29, 37, 53, 59, 67, 83, 101, 107, 109, 131, 139, 149, 163, 173, 179, 181, 197, 227, 229, 269, 277, ... | A019374 |
60 | 19, 23, 29, 41, 47, 73, 79, 83, 89, 97, 107, 149, 157, 167, 227, 263, 269, 313, 331, 337, 347, 373, 379, 383, ... | A019384 |
64 | None | |
72 | 5, 11, 13, 19, 29, 37, 43, 53, 59, 83, 101, 107, 109, 131, 139, 149, 157, 163, 173, 179, 197, 211, 227, 229, ... | A019395 |
80 | 3, 7, 13, 17, 23, 47, 53, 67, 83, 97, 107, 113, 137, 157, 163, 167, 193, 197, 223, 227, 233, 257, 263, 277, 283, ... | A019403 |
90 | 11, 17, 19, 23, 29, 47, 59, 97, 101, 113, 131, 137, 139, 149, 167, 179, 181, 193, 223, 229, 233, 251, 257, 263, ... | A019412 |
96 | 7, 11, 17, 31, 37, 59, 61, 83, 89, 103, 107, 109, 113, 127, 131, 151, 157, 179, 181, 223, 227, 229, 233, 251, ... | A019418 |
100 | None |
The smallest full-reptend primes in base n are:
- 2, 3, 2, 0, 2, 11, 2, 3, 2, 7, 2, 5, 2, 3, 2, 0, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 11, 2, 3, 2, 19, 2, 0, 2, 3, 2, 7, 2, 5, 2, 3, 2, 11, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 19, 2, 3, 2, 0, 2, 7, 2, 3, 2, 19, 2, 5, ... (sequence A056619 in OEIS)
Applications to cryptography[edit]
Binary full reptend prime sequences (also called maximum-length decimal sequences) have found cryptographic and error-correction coding applications. [2] In these applications repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for (when 2 is a primitive root of p) is given by:[3]
These sequences of period p-1 have an autocorrelation function that has a negative peak of -1 for shift of (p-1)/2. The randomness of these sequences has been examined by diehard tests.[4]
See also[edit]
References[edit]
- ^ a b Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
- ^ Kak, Subhash, Chatterjee, A. "On decimal sequences." IEEE Transactions on Information Theory, vol. IT-27, pp. 647-652, September 1981.
- ^ Kak, Subhash, "Encryption and error-correction using d-sequences." IEEE Trans. On Computers, vol. C-34, pp. 803-809, 1985.
- ^ Bellamy, J. "Randomness of D sequences via diehard testing." 2013. arXiv:1312.3618
- Weisstein, Eric W., "Artin's Constant", MathWorld.
- Weisstein, Eric W., "Full Reptend Prime", MathWorld.
- Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.
- Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers"; in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240–246.
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