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| In [[number theory]], a '''full reptend prime''' or '''long prime''' in [[Radix|base]] ''b'' is a [[prime number]] ''p'' such that the formula
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| :<math>\frac{b^{p - 1} - 1}{p}</math>
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| (where ''p'' does not [[Divisor|divide]] ''b'') gives a [[cyclic number]]. Therefore the digital expansion of <math>1/p</math> in base ''b'' repeats the digits of the corresponding cyclic number infinitely. [[Decimal|Base 10]] may be assumed if no base is specified.
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| The first few values of ''p'' for which this formula produces cyclic numbers in decimal are {{OEIS|id=A001913}}
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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 …
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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 0.142857142857142857142857...
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| 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).
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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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| The term "long prime" was used by [[John Horton Conway|John Conway]] and [[Richard Guy]] in their ''Book of Numbers''. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers."
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| The corresponding cyclic number to prime ''p'' will possess ''p'' - 1 digits [[if and only if]] ''p'' is a full reptend prime.
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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 style="-moz-column-count:2; column-count:2;">
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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>
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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"
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| !align="center"|40''k''+1
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| !align="center"|40''k''+3
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| !align="center"|40''k''+9
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| !align="center"|40''k''+13
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| !align="center"|40''k''+27
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| !align="center"|40''k''+31
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| !align="center"|40''k''+37
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| !align="center"|40''k''+39
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| |-
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| |align="center"|41<br>period 5
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| |align="center"|43<br>period 21
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| |align="center"|89<br>period 44
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| |align="center"|13<br>period 6
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| |align="center"|67<br>period 33
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| |align="center"|31<br>period 15
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| |align="center"|37<br>period 3
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| |align="center"|79<br>period 13
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| |-
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| |align="center"|241<br>period 30
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| |align="center"|83<br>period 41
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| |align="center"|409<br>period 204
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| |align="center"|53<br>period 13
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| |align="center"|107<br>period 53
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| |align="center"|71<br>period 35
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| |align="center"|157<br>period 78
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| |align="center"|199<br>period 99
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| |-
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| |align="center"|281<br>period 28
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| |align="center"|163<br>period 81
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| |align="center"|449<br>period 32
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| |align="center"|173<br>period 43
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| |align="center"|227<br>period 113
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| |align="center"|151<br>period 75
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| |align="center"|197<br>period 98
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| |align="center"|239<br>period 7
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| |-
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| |align="center"|401<br>period 200
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| |align="center"|283<br>period 141
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| |align="center"|569<br>period 284
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| |align="center"|293<br>period 146
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| |align="center"|307<br>period 153
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| |align="center"|191<br>period 95
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| |align="center"|277<br>period 69
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| |align="center"|359<br>period 179
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| |-
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| |}
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| However, studies show that ''two-thirds'' of primes of the form 40''k''+''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 120''k''+23 below 100000 are full reptend primes, with 20903 being the first that is not full reptend.
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| ==[[Base 2]] Full reptend primes==
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| 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......({{OEIS|id=A001122}})
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| all of them are 8k+3 or 8k+5.
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| ==References==
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| *{{MathWorld|urlname=ArtinsConstant|title=Artin's Constant}}
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| *{{MathWorld|urlname=FullReptendPrime|title=Full Reptend Prime}}
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| *[[John Horton Conway|Conway, J. H.]] and [[Richard K. Guy|Guy, R. K]]. The Book of Numbers. New York: Springer-Verlag, 1996.
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| *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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| [[Category:Classes of prime numbers]]
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