Full reptend prime
In number theory, a full reptend prime, full repetend prime, proper prime or long prime in base b is an odd prime number p such that the Fermat quotient
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, the multiplicative order ordp b = p − 1, which is equivalent to b being a primitive root modulo p.
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."
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. 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: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 | 3 period 1 | 89 period 44 | 13 period 6 | 67 period 33 | 31 period 15 | 37 period 3 | 79 period 13 |
| 241 period 30 | 43 period 21 | 409 period 204 | 53 period 13 | 107 period 53 | 71 period 35 | 157 period 78 | 199 period 99 |
| 281 period 28 | 83 period 41 | 449 period 32 | 173 period 43 | 227 period 113 | 151 period 75 | 197 period 98 | 239 period 7 |
| 401 period 200 | 163 period 81 | 569 period 284 | 293 period 146 | 307 period 153 | 191 period 95 | 277 period 69 | 359 period 179 |
| 521 period 52 | 283 period 141 | 769 period 192 | 373 period 186 | 347 period 173 | 271 period 5 | 317 period 79 | 439 period 219 |
| 601 period 300 | 443 period 221 | 809 period 202 | 613 period 51 | 467 period 233 | 311 period 155 | 397 period 99 | 479 period 239 |
However, studies show that two-thirds of primes of the form 40k + n, where n ∈ 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:For these primes, 2 is a primitive root modulo p, so 2n modulo p can be any natural number between 1 and p − 1.
These sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of. The randomness of these sequences has been examined by diehard tests.
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, and the period of in base 2 must divide and cannot be p − 1, so they are not full reptend primes in base 2.
Further, all safe primes congruent to 3 are full reptend primes in base 2. For example, 3, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, etc.
Binary full reptend prime sequences have found cryptographic and error-correction coding applications. In these applications, repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for is given by:
The following is a list about the periods to the primes congruent to 1 or 7 :
| 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 |
None of them are binary full reptend primes.
The binary period of nth prime are
The binary period level of nth prime are
However, studies show that three-fourths of primes of the form 8k+n, where n ∈ are full reptend primes in base 2. For some sequences, the preponderance of full reptend primes is much greater. For instance, 1078 of the 1206 primes of form 24k+5 below 100000 are full reptend primes in base 2, with 1013 being the first that is not full reptend in base 2.
''n''-th level reptend prime
An n-th level reptend prime is a prime p having n different cycles in expansions of . In base 10, smallest n-th level reptend prime areIn base 2, smallest n-th level reptend prime are
| n | n-th level reptend primes | OEIS sequence |
| 1 | 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,... | |
| 2 | 3, 13, 31, 43, 67, 71, 83, 89, 107, 151, 157, 163, 191, 197, 199, 227, 283, 293, 307, 311, 347, 359, 373, 401, 409, 431, 439, 443, 467, 479, 523, 557, 563, 569, 587, 599,... | |
| 3 | 103, 127, 139, 331, 349, 421, 457, 463, 607, 661, 673, 691, 739, 829, 967, 1657, 1669, 1699, 1753, 1993, 2011, 2131, 2287, 2647, 2659, 2749, 2953, 3217, 3229, 3583, 3691, 3697, 3739, 3793, 3823, 3931,... | |
| 4 | 53, 173, 277, 317, 397, 769, 773, 797, 809, 853, 1009, 1013, 1093, 1493, 1613, 1637, 1693, 1721, 2129, 2213, 2333, 2477, 2521, 2557, 2729, 2797, 2837, 3329, 3373, 3517, 3637, 3733, 3797, 3853, 3877,... | |
| 5 | 11, 251, 1061, 1451, 1901, 1931, 2381, 3181, 3491, 3851, 4621, 4861, 5261, 6101, 6491, 6581, 6781, 7331, 8101, 9941, 10331, 10771, 11251, 11261, 11411, 12301, 14051, 14221, 14411,... | |
| 6 | 79, 547, 643, 751, 907, 997, 1201, 1213, 1237, 1249, 1483, 1489, 1627, 1723, 1747, 1831, 1879, 1987, 2053, 2551, 2683, 3049, 3253, 3319, 3613, 3919, 4159, 4507, 4519, 4801, 4813, 4831, 4969,... | |
| 7 | 211, 617, 1499, 2087, 2857, 6007, 6469, 7127, 7211, 7589, 9661, 10193, 13259, 13553, 14771, 18047, 18257, 19937, 20903, 21379, 23549, 26153, 27259, 27539, 32299, 33181, 33461, 34847, 35491, 35897,... | |
| 8 | 41, 241, 1601, 1609, 2441, 2969, 3041, 3449, 3929, 4001, 4409, 5009, 6089, 6521, 6841, 8161, 8329, 8609, 9001, 9041, 9929, 13001, 13241, 14081, 14929, 16001, 16481, 17489, 17881, 18121, 19001,... | |
| 9 | 73, 1423, 1459, 2377, 2503, 3457, 7741, 9433, 10891, 10909, 16057, 17299, 17623, 20269, 21313, 22699, 24103, 26263, 28621, 28927, 29629, 30817, 32257, 34273, 34327,... | |
| 10 | 281, 521, 1031, 1951, 2281, 2311, 2591, 3671, 5471, 5711, 6791, 7481, 8111, 8681, 8761, 9281, 9551, 10601, 11321, 12401, 13151, 13591, 14831, 14951, 15671, 16111, 16361, 18671,... | |
| n | n-th level reptend primes | OEIS sequence |
| 1 | 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,... | |
| 2 | 7, 17, 23, 41, 47, 71, 79, 97, 103, 137, 167, 191, 193, 199, 239, 263, 271, 311, 313, 359, 367, 383, 401, 409, 449, 463, 479, 487, 503, 521, 569, 599, 607, 647, 719, 743, 751, 761, 769,... | |
| 3 | 43, 109, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2749, 2917,... | |
| 4 | 113, 281, 353, 577, 593, 617, 1033, 1049, 1097, 1153, 1193, 1201, 1481, 1601, 1889, 2129, 2273, 2393, 2473, 3049, 3089, 3137, 3217, 3313, 3529, 3673, 3833, 4001, 4217, 4289, 4457, 4801, 4817, 4937,... | |
| 5 | 251, 571, 971, 1181, 1811, 2011, 2381, 2411, 3221, 3251, 3301, 3821, 4211, 4861, 4931, 5021, 5381, 5861, 6221, 6571, 6581, 8461, 8501, 9091, 9461, 10061, 10211, 10781, 11251, 11701, 11941, 12541,... | |
| 6 | 31, 223, 433, 439, 457, 727, 919, 1327, 1399, 1423, 1471, 1831, 1999, 2017, 2287, 2383, 2671, 2767, 2791, 2953, 3271, 3343, 3457, 3463, 3607, 3631, 3823, 3889, 4129, 4423, 4519, 4567, 4663, 4729, 4759,... | |
| 7 | 1163, 1709, 2003, 3109, 3389, 3739, 5237, 5531, 5867, 7309, 9157, 9829, 10627, 10739, 11117, 11243, 11299, 11411, 11467, 13259, 18803, 20147, 20483, 21323, 21757, 27749, 27763, 29947,... | |
| 8 | 73, 89, 233, 937, 1217, 1249, 1289, 1433, 1553, 1609, 1721, 1913, 2441, 2969, 3257, 3449, 4049, 4201, 4273, 4297, 4409, 4481, 4993, 5081, 5297, 5689, 6089, 6449, 6481, 6689, 6857, 7121, 7529, 7993,... | |
| 9 | 397, 7867, 10243, 10333, 12853, 13789, 14149, 14293, 14563, 15643, 17659, 18379, 18541, 21277, 21997, 23059, 23203, 26731, 27739, 29179, 29683, 31771, 34147, 35461, 35803, 36541, 37747, 39979,... | |
| 10 | 151, 241, 431, 641, 911, 3881, 4751, 4871, 5441, 5471, 5641, 5711, 6791, 6871, 8831, 9041, 9431, 10711, 12721, 13751, 14071, 14431, 14591, 15551, 16631, 16871, 17231, 17681, 17791, 18401, 19031, 19471,... |
Full reptend primes in various bases
Artin also conjectured:- There are infinitely many full-reptend primes in all bases except squares.
- Full-reptend primes in all bases except perfect powers and numbers whose squarefree part are congruent to 1 to mod 4 comprise 37.395...% of all primes.
| Base | Full reptend primes | OEIS sequence |
| −30 | 7, 41, 61, 83, 89, 107, 109, 127, 139, 173, 193, 197, 211, 227, 239, 281, 293, 311, 317, 331, 347, 349, 359,... | |
| −29 | 2, 17, 23, 41, 59, 71, 73, 83, 89, 97, 101, 103, 107, 113, 137, 139, 167, 179, 199, 223, 227, 229, 239, 269,... | |
| −28 | 3, 5, 13, 17, 19, 31, 41, 47, 59, 73, 83, 89, 101, 103, 131, 139, 167, 173, 181, 227, 229, 251, 257, 269, 283,... | |
| −27 | 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,... | |
| −26 | 11, 23, 29, 41, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 127, 137, 157, 163, 173, 191, 193, 199, 227, 263,... | |
| −25 | 2, 3, 7, 11, 19, 23, 43, 47, 59, 79, 83, 103, 107, 131, 139, 151, 167, 179, 223, 227, 239, 263, 283, 307, 311,... | |
| −24 | 13, 17, 19, 37, 41, 43, 47, 71, 89, 109, 113, 137, 139, 157, 163, 167, 181, 191, 211, 229, 233, 257, 263, 277,... | |
| −23 | 2, 5, 7, 17, 19, 43, 67, 83, 89, 97, 107, 113, 137, 149, 181, 191, 199, 227, 229, 251, 263, 281, 283, 293, 337,... | |
| −22 | 3, 5, 17, 37, 41, 53, 59, 151, 167, 179, 193, 233, 251, 263, 269, 271, 281, 317, 337, 359, 379, 389, 397, 409,... | |
| −21 | 2, 29, 47, 53, 59, 67, 83, 97, 113, 127, 131, 137, 149, 151, 157, 167, 181, 197, 227, 233, 251, 281, 311, 313,... | |
| −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,... | |
| −19 | 2, 3, 13, 29, 31, 37, 41, 53, 59, 67, 71, 79, 89, 103, 107, 113, 167, 173, 179, 193, 223, 227, 257, 269, 281,... | |
| −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,... | |
| −17 | 2, 5, 19, 37, 41, 43, 47, 59, 61, 67, 83, 97, 103, 113, 127, 151, 173, 179, 191, 193, 197, 233, 239, 251, 263,... | |
| −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,... | |
| −15 | 2, 11, 13, 29, 37, 41, 43, 59, 71, 73, 89, 97, 101, 103, 127, 131, 149, 157, 163, 179, 191, 193, 239, 251, 269,... | |
| −14 | 11, 17, 29, 31, 43, 47, 53, 73, 89, 97, 107, 109, 149, 163, 167, 179, 199, 241, 257, 271, 277, 311, 313, 317,... | |
| −13 | 2, 3, 5, 23, 37, 41, 43, 73, 79, 89, 97, 107, 109, 127, 131, 137, 139, 149, 179, 191, 197, 199, 241, 251, 263,... | |
| −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,... | |
| −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,... | |
| −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,... | |
| −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,... | |
| −8 | 5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 461, 479,... | |
| −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,... | |
| −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,... | |
| −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,... | |
| −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,... | |
| −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,... | |
| −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,... | |
| 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,... | |
| 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,... | |
| 4 | ||
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 16 | ||
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... |
The smallest full-reptend primes in base n are: