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In [[number theory]], a '''full reptend prime''', '''full repetend prime''', '''proper prime'''<ref name= Dickson>Dickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co.</ref>{{rp|166}} or '''long prime''' in [[Radix|base]] ''b'' is an [[parity (mathematics)|odd]] [[prime number]] ''p'' such that the [[Fermat quotient]] |
In [[number theory]], a '''full reptend prime''', '''full repetend prime''', '''proper prime'''<ref name= Dickson>Dickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co.</ref>{{rp|166}} or '''long prime''' in [[Radix|base]] ''b'' is an [[parity (mathematics)|odd]] [[prime number]] ''p'' such that the [[Fermat quotient]] |
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:<math>q_p(b) = \frac{b^{p - 1} - 1}{p}</math> |
: <math>q_p(b) = \frac{b^{p - 1} - 1}{p}</math> |
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(where ''p'' does not [[Divisor|divide]] ''b'') gives a [[cyclic number]]. Therefore, the base ''b'' expansion of <math>1/p</math> repeats the digits of the corresponding cyclic number infinitely, as does that of <math>a/p</math> with rotation of the digits for any ''a'' between 1 and ''p'' |
(where ''p'' does not [[Divisor|divide]] ''b'') gives a [[cyclic number]]. Therefore, the base ''b'' expansion of <math>1/p</math> repeats the digits of the corresponding cyclic number infinitely, as does that of <math>a/p</math> 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]] {{nobr|ord<sub>''p''</sub> ''b''}} = ''p'' − 1, which is equivalent to ''b'' being a [[primitive root modulo n|primitive root]] modulo ''p''. |
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The term "long prime" was used by [[John Horton Conway|John Conway]] and [[Richard K. Guy|Richard Guy]] in their ''Book of Numbers''. Confusingly, [[Neil Sloane|Sloane]]'s [[OEIS]] refers to these primes as "cyclic numbers |
The term "long prime" was used by [[John Horton Conway|John Conway]] and [[Richard K. Guy|Richard Guy]] in their ''Book of Numbers''. Confusingly, [[Neil Sloane|Sloane]]'s [[OEIS]] refers to these primes as "cyclic numbers". |
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== Base 10 == |
== Base 10 == |
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[[Decimal|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 reptend the same number of times as each other digit.<ref name=Dickson/>{{rp|166}} (For such primes in base 10, see {{oeis|id=A073761}}. In fact, in base ''b'', if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., ''b'' |
[[Decimal|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 reptend the same number of times as each other digit.<ref name=Dickson/>{{rp|166}} (For such primes in base 10, see {{oeis|id=A073761}}.) In fact, in base ''b'', if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., ''b'' − 1 appears in the repetend the same number of times as each other digit, but no such prime exists when ''b'' = 12, since every full reptend prime in [[duodecimal|base 12]] ends in the digit 5 or 7 in the same base. Generally, no such prime exists when ''b'' is [[Modular arithmetic|congruent]] to 0 or 1 modulo 4. |
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The values of ''p'' |
The values of ''p'' for which this formula produces cyclic numbers in decimal are: |
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:[[7 (number)|7]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[47 (number)|47]], [[59 (number)|59]], [[61 (number)|61]], [[97 (number)|97]], [[109 (number)|109]], [[113 (number)|113]], [[131 (number)|131]], [[149 (number)|149]], [[167 (number)|167]], [[179 (number)|179]], [[181 (number)|181]], [[193 (number)|193]], [[223 (number)|223]], [[229 (number)|229]], [[233 (number)|233]], [[257 (number)|257]], [[263 (number)|263]], [[269 (number)|269]], [[313 (number)|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, ... {{OEIS|id=A001913}} |
: [[7 (number)|7]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[47 (number)|47]], [[59 (number)|59]], [[61 (number)|61]], [[97 (number)|97]], [[109 (number)|109]], [[113 (number)|113]], [[131 (number)|131]], [[149 (number)|149]], [[167 (number)|167]], [[179 (number)|179]], [[181 (number)|181]], [[193 (number)|193]], [[223 (number)|223]], [[229 (number)|229]], [[233 (number)|233]], [[257 (number)|257]], [[263 (number)|263]], [[269 (number)|269]], [[313 (number)|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, 1019, 1021, 1033, 1051... {{OEIS|id=A001913}} |
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This sequence is the set of primes ''p'' such that 10 is a [[primitive root modulo n|primitive root]] modulo ''p''. [[Artin's conjecture on primitive roots]] is that this sequence contains 37.395...% of the primes. |
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For example, the case ''b'' = 10, ''p'' = 7 gives the cyclic number [[142857 (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... |
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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. |
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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 n|primitive root]] modulo ''p''. [[Artin's conjecture on primitive roots]] is that this sequence contains 37.395..% of the primes. |
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==Patterns of occurrence of full reptend primes== |
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Advanced [[modular arithmetic]] can show that any prime of the following forms: |
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{{div col|colwidth=15em}} |
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#40''k'' + 1 |
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#40''k'' + 3 |
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#40''k'' + 9 |
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#40''k'' + 13 |
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#40''k'' + 27 |
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#40''k'' + 31 |
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#40''k'' + 37 |
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#40''k'' + 39 |
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{{div col end}} |
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can ''never'' be a full reptend prime in base 10. The first primes of these forms, with their periods, are: |
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{| class="wikitable" style="text-align:center" |
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!40''k'' + 1 |
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!40''k'' + 3 |
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!40''k'' + 9 |
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!40''k'' + 13 |
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!40''k'' + 27 |
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!40''k'' + 31 |
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!40''k'' + 37 |
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!40''k'' + 39 |
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|- |
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|41<br>period 5 |
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|3<br>period 1 |
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|89<br>period 44 |
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|13<br>period 6 |
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|67<br>period 33 |
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|31<br>period 15 |
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|37<br>period 3 |
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|79<br>period 13 |
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|- |
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|241<br>period 30 |
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|43<br>period 21 |
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|409<br>period 204 |
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|53<br>period 13 |
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|107<br>period 53 |
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|71<br>period 35 |
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|157<br>period 78 |
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|199<br>period 99 |
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|- |
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|281<br>period 28 |
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|83<br>period 41 |
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|449<br>period 32 |
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|173<br>period 43 |
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|227<br>period 113 |
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|151<br>period 75 |
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|197<br>period 98 |
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|239<br>period 7 |
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|- |
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|401<br>period 200 |
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|163<br>period 81 |
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|569<br>period 284 |
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|293<br>period 146 |
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|307<br>period 153 |
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|191<br>period 95 |
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|277<br>period 69 |
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|359<br>period 179 |
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|- |
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|521<br>period 52 |
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|283<br>period 141 |
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|769<br>period 192 |
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|373<br>period 186 |
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|347<br>period 173 |
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|271<br>period 5 |
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|317<br>period 79 |
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|439<br>period 219 |
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|- |
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|601<br>period 300 |
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|443<br>period 221 |
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|809<br>period 202 |
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|613<br>period 51 |
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|467<br>period 233 |
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|311<br>period 155 |
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|397<br>period 99 |
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|479<br>period 239 |
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|} |
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However, studies show that ''two-thirds'' of primes of the form 40''k'' + ''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 120''k'' + 23 below 100000 are full reptend primes, with 20903 being the first that is not full reptend. |
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==Binary full reptend primes== |
==Binary full reptend primes== |
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| Line 106: | Line 21: | ||
: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, ... {{OEIS|id=A001122}} |
: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, ... {{OEIS|id=A001122}} |
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For these primes, 2 is a [[primitive root modulo n|primitive root]] modulo ''p'', so 2<sup>''n''</sup> [[modulo operation|modulo]] ''p'' can be any [[natural number]] between 1 and ''p'' |
For these primes, 2 is a [[primitive root modulo n|primitive root]] modulo ''p'', so 2<sup>''n''</sup> [[modulo operation|modulo]] ''p'' can be any [[natural number]] between 1 and ''p'' − 1. |
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:<math>a(i) = 2^{i}~\bmod p ~\bmod 2</math> |
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These sequences of period ''p'' − 1 have an autocorrelation function that has a negative peak of −1 for shift of <math>(p-1)/2</math>. The randomness of these sequences has been examined by [[diehard tests]].<ref>Bellamy, J. "Randomness of D sequences via diehard testing." 2013. {{arXiv|1312.3618}}</ref> |
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All of them are of form 8''k'' + 3 or 8''k'' + 5, because if ''p'' = 8''k'' + 1 or 8''k'' + 7, then 2 is a [[quadratic residue]] modulo ''p'', so ''p'' divides <math>2^{(p - 1)/2} - 1</math>, and the period of <math>1/p</math> in base 2 must divide <math>(p - 1)/2</math> and cannot be ''p'' − 1, so they are not full reptend primes in base 2. |
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Further, all [[safe prime]]s congruent to 3 modulo 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, 1523, 1619, 1907, etc. (less than 2000) |
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Binary full reptend prime sequences (also called maximum-length decimal sequences) have found [[cryptographic]] and [[error-correction code|error-correction coding]] applications.<ref>Kak, Subhash, Chatterjee, A. "On decimal sequences." IEEE Transactions on Information Theory, vol. IT-27, pp. 647-652, September 1981.</ref> In these applications, repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for <math>1/p</math> (when 2 is a primitive root of ''p'') is given by:<ref>Kak, Subhash, "Encryption and error-correction using d-sequences." IEEE Trans. On Computers, vol. C-34, pp. 803-809, 1985.</ref> |
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The following is a list about the periods (in binary) to the primes congruent to 1 or 7 (mod 8): (less than 1000) |
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{| class="wikitable" style="text-align:right" |
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!8''k'' + 1 |
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|17 |
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|41 |
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|73 |
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|89 |
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|97 |
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|113 |
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|137 |
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|193 |
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|233 |
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|241 |
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|257 |
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|281 |
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|313 |
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|337 |
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|353 |
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|401 |
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|409 |
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|433 |
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|449 |
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|457 |
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|521 |
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|569 |
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|- |
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!period |
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|8 |
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|20 |
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|9 |
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|11 |
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|48 |
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|28 |
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|68 |
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|96 |
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|29 |
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|24 |
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|16 |
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|70 |
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|156 |
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|21 |
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|88 |
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|200 |
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|204 |
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|72 |
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|224 |
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|76 |
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|260 |
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|284 |
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|- |
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!8''k'' + 1 |
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|577 |
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|593 |
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|601 |
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|617 |
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|641 |
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|673 |
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|761 |
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|769 |
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|809 |
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|857 |
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|881 |
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|929 |
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|937 |
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|953 |
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|977 |
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|1009 |
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|1033 |
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|1049 |
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|1097 |
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|1129 |
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|1153 |
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|1193 |
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|- |
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!period |
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|144 |
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|148 |
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|25 |
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|154 |
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|64 |
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|48 |
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|380 |
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|384 |
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|404 |
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|428 |
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|55 |
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|464 |
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|117 |
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|68 |
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|488 |
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|504 |
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|258 |
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|262 |
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|274 |
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|564 |
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|288 |
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|298 |
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|- |
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!8''k'' + 7 |
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|7 |
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|23 |
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|31 |
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|47 |
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|71 |
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|79 |
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|103 |
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|127 |
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|151 |
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|167 |
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|191 |
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|199 |
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|223 |
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|239 |
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|263 |
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|271 |
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|311 |
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|359 |
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|367 |
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|383 |
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|431 |
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|439 |
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|- |
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!period |
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|3 |
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|11 |
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|5 |
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|23 |
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|35 |
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|39 |
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|51 |
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|7 |
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|15 |
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|83 |
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|95 |
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|99 |
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|37 |
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|119 |
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|131 |
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|135 |
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|155 |
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|179 |
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|183 |
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|191 |
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|43 |
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|73 |
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|- |
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!8''k'' + 7 |
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|463 |
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|479 |
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|487 |
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|503 |
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|599 |
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|607 |
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|631 |
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|647 |
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|719 |
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|727 |
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|743 |
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|751 |
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|823 |
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|839 |
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|863 |
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|887 |
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|911 |
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|919 |
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|967 |
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|983 |
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|991 |
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|1031 |
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|- |
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!period |
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|231 |
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|239 |
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|243 |
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|251 |
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|299 |
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|303 |
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|45 |
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|323 |
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|359 |
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|121 |
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|371 |
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|375 |
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|411 |
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|419 |
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|431 |
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|443 |
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|91 |
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|153 |
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|483 |
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|491 |
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|495 |
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|515 |
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|} |
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''None'' of them are binary full reptend primes. |
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The binary period of ''n''th prime are |
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: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) {{OEIS|id=A014664}} |
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The binary period level of ''n''th prime are |
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: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, ... {{OEIS|id=A001917}} |
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However, studies show that ''three-fourths'' of primes of the form 8''k'' + ''n'', where ''n'' ∈ {3, 5} are full reptend primes in base 2 (For example, there are 87 primes below 1000 congruent to 3 or 5 modulo 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, 1078 of the 1206 primes of form 24''k'' + 5 below 100000 are full reptend primes in base 2, with 1013 being the first that is not full reptend in base 2. |
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==''n''-th level reptend prime== |
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An '''''n''-th level reptend prime''' is a prime ''p'' having ''n'' different cycles in expansions of <math>\frac{k}{p}</math> (''k'' is an [[integer]], 1 ≤ ''k'' ≤ ''p''−1). In base 10, smallest ''n''-th level reptend prime are |
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:7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 859, 239, 27581, 9613, 18131, 13757, 33931, 9161, 118901, 6763, 18233, 1409, 88741, 4003, 5171, 19489, 86143, 23201, ... {{OEIS|id=A054471}} |
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In base 2, smallest ''n''-th level reptend prime are |
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:3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593, 17029, 2687, 42701, 11161, 13099, 1103, 71293, 13121, 17467, 2143, 83077, 25609, 5581, 5153, 26227, 2113, 51941, 2351, ... {{OEIS|id=A101208}} |
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{|class="wikitable" |
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!''n'' |
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!''n''-th level reptend primes (in decimal) |
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![[OEIS]] sequence |
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|- |
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|1 |
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|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, ... |
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|{{OEIS link|id=A006883}} |
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|- |
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|2 |
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|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, ... |
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|{{OEIS link|id=A275081}} |
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|- |
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|3 |
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|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, ... |
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|{{OEIS link|id=A055628}} |
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|- |
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|4 |
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|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, ... |
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|{{OEIS link|id=A056157}} |
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|- |
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|5 |
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|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, ... |
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|{{OEIS link|id=A056210}} |
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|- |
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|6 |
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|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, ... |
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|{{OEIS link|id=A056211}} |
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|- |
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|7 |
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|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, ... |
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|{{OEIS link|id=A056212}} |
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|- |
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|8 |
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|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, ... |
|||
|{{OEIS link|id=A056213}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A056214}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A056215}} |
|||
|- |
|||
!''n'' |
|||
!''n''-th level reptend primes (in binary) |
|||
![[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, ... |
|||
|{{OEIS link|id=A001122}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A115591}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A001133}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A001134}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A001135}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A001136}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A152307}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A152308}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A152309}} |
|||
|- |
|||
|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, ... |
|||
|{{OEIS link|id=A152310}} |
|||
|} |
|||
==Full reptend primes in various bases== |
|||
Artin also [[conjecture]]d: |
|||
* There are infinitely many full-reptend primes in all bases except [[square number|square]]s. |
|||
* Full-reptend primes in all bases except [[perfect power]]s and numbers whose [[squarefree]] part are congruent to 1 modulo 4 comprise 37.395...% of all primes. (See {{oeis|id=A085397}}) |
|||
: <math>a(i) = 2^i \bmod p \bmod 2.</math> |
|||
{|class="wikitable" |
|||
These sequences of period ''p'' − 1 have an autocorrelation function that has a negative peak of −1 for shift of <math>(p-1)/2</math>. The randomness of these sequences has been examined by [[diehard tests]].<ref>Bellamy, J. "Randomness of D sequences via diehard testing". 2013. {{arXiv|1312.3618}}.</ref> |
|||
!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, ... |
|||
|{{OEIS link|id=A105902}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105901}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105900}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105875}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105898}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105897}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105896}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105895}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105894}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105893}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105892}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105891}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105890}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105889}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105876}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105887}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105886}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=A105885}} |
|||
|- |
|||
|−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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=A105883}} |
|||
|- |
|||
|[[Base -10|−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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=A105876}} |
|||
|- |
|||
|[[Base -3|−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, ... |
|||
|{{OEIS link|id=A105875}} |
|||
|- |
|||
|[[Base -2|−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, ... |
|||
|{{OEIS link|id=A105874}} |
|||
|- |
|||
|[[Base 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, ... |
|||
|{{OEIS link|id=A001122}} |
|||
|- |
|||
|[[Base 3|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, ... |
|||
|{{OEIS link|id=A019334}} |
|||
|- |
|||
|[[Base 4|4]] |
|||
|(none) |
|||
| |
|||
|- |
|||
|[[Base 5|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, ... |
|||
|{{OEIS link|id=A019335}} |
|||
|- |
|||
|[[Base 6|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, ... |
|||
|{{OEIS link|id=A019336}} |
|||
|- |
|||
|[[Base 7|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, ... |
|||
|{{OEIS link|id=A019337}} |
|||
|- |
|||
|[[Base 8|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, ... |
|||
|{{OEIS link|id=A019338}} |
|||
|- |
|||
|[[Base 9|9]] |
|||
|2 (no others) |
|||
| |
|||
|- |
|||
|[[Base 10|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, ... |
|||
|{{OEIS link|id=A001913}} |
|||
|- |
|||
|[[Base 11|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, ... |
|||
|{{OEIS link|id=A019339}} |
|||
|- |
|||
|[[Duodecimal|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, ... |
|||
|{{OEIS link|id=A019340}} |
|||
|- |
|||
|[[Base 13|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, ... |
|||
|{{OEIS link|id=A019341}} |
|||
|- |
|||
|[[Base 14|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, ... |
|||
|{{OEIS link|id=A019342}} |
|||
|- |
|||
|[[Base 15|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, ... |
|||
|{{OEIS link|id=A019343}} |
|||
|- |
|||
|[[Base 16|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, ... |
|||
|{{OEIS link|id=A019344}} |
|||
|- |
|||
|[[Base 18|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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=A019346}} |
|||
|- |
|||
|[[Base 20|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, ... |
|||
|{{OEIS link|id=A019347}} |
|||
|- |
|||
|[[Base 21|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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=A019350}} |
|||
|- |
|||
|[[Base 24|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, ... |
|||
|{{OEIS link|id=A019351}} |
|||
|- |
|||
|[[Base 25|25]] |
|||
|2 (no others) |
|||
| |
|||
|- |
|||
|[[Base26|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, ... |
|||
|{{OEIS link|id=A019352}} |
|||
|- |
|||
|[[Base 27|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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=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, ... |
|||
|{{OEIS link|id=A019356}} |
|||
|} |
|||
Binary full reptend prime sequences (also called maximum-length decimal sequences) have found [[cryptographic]] and [[error-correction code|error-correction coding]] applications.<ref>Kak, Subhash, Chatterjee, A. "On decimal sequences". IEEE Transactions on Information Theory, vol. IT-27, pp. 647–652, September 1981.</ref> In these applications, repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for <math>1/p</math> (when 2 is a primitive root of ''p'') is given by Kak.<ref>Kak, Subhash, "Encryption and error-correction using d-sequences". IEEE Trans. On Computers, vol. C-34, pp. 803–809, 1985.</ref> |
|||
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, 2, 3, 2, 13, 2, 5, 2, 3, 2, 5, 2, 11, 2, 3, 2, 5, 2, 11, 2, 3, 2, 7, 2, 7, 2, 3, 2, 0, ... {{OEIS|id=A056619}} |
|||
==See also== |
==See also== |
||
Latest revision as of 01:38, 13 June 2024
In number theory, a full reptend prime, full repetend prime, proper prime[1]: 166 or long prime in base b is an odd prime number p such that the Fermat quotient
(where p does not divide b) gives a cyclic number. Therefore, the base b expansion of 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
[edit]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 reptend the same number of times as each other digit.[1]: 166 (For such primes in base 10, see OEIS: A073761.) In fact, in base b, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., b − 1 appears in the repetend the same number of times as each other digit, but no such prime exists when b = 12, since every full reptend prime in base 12 ends in the digit 5 or 7 in the same base. Generally, no such prime exists when b is congruent to 0 or 1 modulo 4.
The values of p 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, 1019, 1021, 1033, 1051... (sequence A001913 in the OEIS)
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.
Binary 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 the OEIS)
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.[2]
Binary full reptend prime sequences (also called maximum-length decimal sequences) have found cryptographic and error-correction coding applications.[3] 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 Kak.[4]
See also
[edit]References
[edit]- ^ a b Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
- ^ Bellamy, J. "Randomness of D sequences via diehard testing". 2013. arXiv:1312.3618.
- ^ 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.
- 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.