Unique prime
No. of known terms  102 

Conjectured no. of terms  Infinite 
First terms  3, 11, 37, 101 
Largest known term  (10^{270343}1)/9 
OEIS index 

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In recreational number theory, a unique prime or unique period prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equal to the period length of the reciprocal of q, 1 / q.^{[1]} For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. In contrast, 41 and 271 both have period 5; 7 and 13 both have period 6; 239 and 4649 both have period 7; 73 and 137 both have period 8; 21649 and 513239 both have period 11; 53, 79 and 265371653 all have period 13; 31 and 2906161 both have period 15; 17 and 5882353 both have period 16; 2071723 and 5363222357 both have period 17; 19 and 52579 both have period 18; 3541 and 27961 both have period 20. Therefore, none of these is a unique prime. Unique primes were first described by Samuel Yates in 1980.
The above definition is related to the decimal representation of integers. Unique primes may be defined and have been studied in any numeral base.
Contents
Period of a prime in base b[edit]
The representation of the reciprocal of a prime number (or, more generally, an integer) p in the numeral base b is periodic of period n if
where q is a positive integer smaller than According to the summation formula of geometric series, this may be rewritten as
In other words, n is a period of the representation of 1/p if and only if p is a divisor of Euler's theorem asserts that, if an integer b is coprime with p, then p is a divisor of where is Euler's totient function. This proves that, for every integer p coprime with b, the representation of the reciprocal of p is periodic in base b.
All the periods of a periodic function are multiples of a shortest period generally called the fundamental period. In this article, we call period of p in base b the shortest period of the representation of 1/p in base b. Therefore, the period of p in base b is the smallest positive integer n such that that p is a divisor of In other words, the period of a prime p in base b is the multiplicative order of b modulo p.
According to Zsigmondy's theorem, every positive integer is a period of some prime in base b except in the following cases:
 b = 2 and n = 1 or 6
 n = 2 and b= 2^{k} − 1 for some integer k > 1
As
where is the nth cyclotomic polynomial, the primes of period n in base b are prime divisors of More precisely, the primes of period n are exactly the prime divisors of that do not divide n (see below for a proof of this result and of the following ones).
If b is even (this includes the binary and the decimal cases), the prime divisors of that do not divide n are exactly the prime divisors of^{[citation needed]}
This is wrong if b is odd: if n = 2 and b = 4k − 1, where k is a positive integer, then
although 2 divides both n = 2 and
If b is odd, the primes of period n are exactly, if n = 1, the prime divisors of , or, if n > 1, the odd prime divisors of R_{n}(b).
Sketch of the proof of the characterization of primes of period n 

As the period of every prime p divides p – 1 (Fermat's little theorem), if p divides n, then its period is smaller than n. Conversely, if p divides and has a period k smaller than n, then it is a common divisor of and As the resultant of two polynomials is a linear combination of these polynomials, p divides the resultant of and As these two polynomials are coprime and divide p divides also the discriminant of Thus, a prime divisor of , that has a period smaller than n, is also a divisor of n. Now, we have to prove that, if a prime p > 2 divides n and then it does not divide In fact, this implies immediately that p does not divide If b is even, 2 cannot divide (which is odd), and the condition p > 2 is not restrictive. Thus, let n = pm. It suffices to prove that does not divides S(b) for some polynomial S(x), which is a multiple of We take By Fermat's little theorem, we have As p divides , we have also Thus the multiplicative order of b modulo p divides gcd(n, p − 1), which is a divisor of m = n/p. Thus c = b^{m} − 1 is a multiple of p. Now, As p is prime and greater than 2, all the terms but the first one are multiple of This proves that does not divides 
A prime p is a unique prime in base b, if and only if, for some n, it is the unique prime divisor of that does not divide n. If b is even (which includes the binary and the decimal cases) this means that
for some positive integer c .
If b is odd, this means that
for some integers c > 0 and d ≥ 0. This provides an efficient method for computing the unique primes and the primes of a given period.
Note that a prime divisor of b is coprime with , and thus also with its divisor Such a prime has no period length, as the representation in base b of its reciprocal is finite instead of being periodic. Thus, such a prime is never considered as a unique prime, even if it is the unique prime that has a finite reciprocal in base b. For example, 2 is not considered as a unique prime in binary, although it is the only prime with finite reciprocal in binary.
Table of the periods of primes up to 139 in bases up to 24  

The mention "terminated" means that the prime divides the base, and thus that the representation of its reciprocal is finite.

Table of primes of a given period (up to 24) in bases up to 24  

Bold for unique primes.

Decimal unique primes[edit]
At present, more than fifty unique primes or probable primes are known. However, there are only twentythree unique primes below 10^{100}. The following table lists all 23 unique primes below 10^{100} (sequence A040017 (sorted) and A007615 (ordered by period length) in OEIS) and their periods (sequence A051627 (ordered by corresponding primes) and A007498 (sorted) in OEIS)
Prime  

1  3 
2  11 
3  37 
4  101 
10  9,091 
12  9,901 
9  333,667 
14  909,091 
24  99,990,001 
36  999,999,000,001 
48  9,999,999,900,000,001 
38  909,090,909,090,909,091 
19  1,111,111,111,111,111,111 
23  11,111,111,111,111,111,111,111 
39  900,900,900,900,990,990,990,991 
62  909,090,909,090,909,090,909,090,909,091 
120  100,009,999,999,899,989,999,000,000,010,001 
150  10,000,099,999,999,989,999,899,999,000,000,000,100,001 
106  9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091 
93  900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991 
134  909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091 
294  142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143 
196  999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001 
The prime with period length 294 is similar^{[clarification needed]} to the reciprocal of 7 (0.142857142857142857...)
Just after the table, the twentyfourth unique prime has 128 digits and period length 320. It can be written as (9_{32}0_{32})_{2} + 1, where a subscript number n indicates n consecutive copies of the digit or group of digits before the subscript.
Though they are rare, based on the occurrence of repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes. (Any repunit prime is unique.)
As of 2010^{[update]} the repunit (10^{270343}1)/9 is the largest known probable unique prime.^{[2]}
In 1996 the largest proven unique prime was (10^{1132} + 1)/10001 or, using the notation above, (99990000)_{141}+ 1. It has 1129 digits. The record has been improved many times since then. As of 2017^{[update]} the largest proven unique prime is , it has 20160 digits.^{[3]}
Binary unique primes[edit]
The first unique primes in binary (base 2) are:
 3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, ... (sequence A144755 (sorted) and A161509 (ordered by period length) in OEIS)
The period length of them are:
 2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, ... (sequence A247071 (ordered by corresponding primes) and A161508 (sorted) in OEIS)
They include Fermat primes (the period length is a power of 2), Mersenne primes (the period length is a prime) and Wagstaff primes (the period length is twice an odd prime).
Additionally, if n is a natural number which is not equal to 1 or 6, than at least one prime have period n in base 2, because of the Zsigmondy theorem. Besides, if n is congruent to 4 (mod 8) and n > 20, then at least two primes have period n in base 2, (Thus, n is not a unique period in base 2) because of the Aurifeuillean factorization, for example, 113 (=) and 29 (=) both have period 28 in base 2, 37 (=) and 109 (=) both have period 36 in base 2, and that 397 (=) and 2113 (=) both have period 44 in base 2,
As shown above, a prime p is a unique prime of period n in base 2 if and only if there exists a natural number c such that
The only known values of n such that is composite but is prime are 18, 20, 21, 54, 147, 342, 602, and 889 (in these case, has a small factor which divides n). It is a conjecture that there is no other n with this property.^{[citation needed]} All other known base 2 unique primes are of the form .
In fact, no prime with c > 1 (that is is a true power of p) have been discovered, and all known unique primes p have c = 1. It is conjectured that all unique primes have c = 1 (that is, all base 2 unique primes are not Wieferich primes).
The largest known base 2 unique prime is 2^{77232917}1, it is also the largest known prime. With an exception of Mersenne primes, the largest known probable base 2 unique prime is ,^{[4]} and the largest proven base 2 unique prime is . Besides, the largest known probable base 2 unique prime which is not Mersenne prime or Wagstaff prime is .
Similar to base 10, though they are rare (but more than the case to base 10), it is conjectured that there are infinitely many base 2 unique primes, because all Mersenne primes are unique in base 2, and it is conjectured they there are infinitely many Mersenne primes.^{[citation needed]}
They divide none of overpseudoprimes to base 2, but every other odd prime number divide one overpseudoprime to base 2, because if and only if a composite number can be written as , it is an overpseudoprime to base 2.^{[clarification needed]}
There are 52 unique primes in base 2 below 2^{64}, they are:
Prime (written in decimal)  Prime (written in binary)  

2  3  11 
4  5  101 
3  7  111 
10  11  1011 
12  13  1101 
8  17  1 0001 
18  19  1 0011 
5  31  1 1111 
20  41  10 1001 
14  43  10 1011 
9  73  100 1001 
7  127  111 1111 
15  151  1001 0111 
24  241  1111 0001 
16  257  1 0000 0001 
30  331  1 0100 1011 
21  337  1 0101 0001 
22  683  10 1010 1011 
26  2,731  1010 1010 1011 
42  5,419  1 0101 0010 1011 
13  8,191  1 1111 1111 1111 
34  43,691  1010 1010 1010 1011 
40  61,681  1111 0000 1111 0001 
32  65,537  1 0000 0000 0000 0001 
54  87,211  1 0101 0100 1010 1011 
17  131,071  1 1111 1111 1111 1111 
38  174,763  10 1010 1010 1010 1011 
27  262,657  100 0000 0010 0000 0001 
19  524,287  111 1111 1111 1111 1111 
33  599,479  1001 0010 0101 1011 0111 
46  2,796,203  10 1010 1010 1010 1010 1011 
56  15,790,321  1111 0000 1111 0000 1111 0001 
90  18,837,001  1 0001 1111 0110 1110 0000 1001 
78  22,366,891  1 0101 0101 0100 1010 1010 1011 
62  715,827,883  10 1010 1010 1010 1010 1010 1010 1011 
31  2,147,483,647  111 1111 1111 1111 1111 1111 1111 1111 
80  4,278,255,361  1111 1111 0000 0000 1111 1111 0000 0001 
120  4,562,284,561  1 0000 1111 1110 1110 1111 0000 0001 0001 
126  77,158,673,929  1 0001 1111 0111 0000 0011 1110 1110 0000 1001 
150  1,133,836,730,401  1 0000 0111 1111 1101 1110 1111 1000 0000 0010 0001 
86  2,932,031,007,403  10 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011 
98  4,363,953,127,297  11 1111 1000 0000 1111 1110 0000 0011 1111 1000 0001 
49  4,432,676,798,593  100 0000 1000 0001 0000 0010 0000 0100 0000 1000 0001 
69  10,052,678,938,039  1001 0010 0100 1001 0010 0101 1011 0110 1101 1011 0111 
65  145,295,143,558,111  1000 0100 0010 0101 0010 1001 0110 1011 0101 1011 1101 1111 
174  96,076,791,871,613,611  1 0101 0101 0101 0101 0101 0101 0100 1010 1010 1010 1010 1010 1010 1011 
77  581,283,643,249,112,959  1000 0001 0001 0010 0010 0110 0100 1100 1101 1001 1011 1011 0111 0111 1111 
93  658,812,288,653,553,079  1001 0010 0100 1001 0010 0100 1001 0011 0110 1101 1011 0110 1101 1011 0111 
122  768,614,336,404,564,651  1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1011 
61  2,305,843,009,213,693,951  1 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 
85  9,520,972,806,333,758,431  1000 0100 0010 0001 0100 1010 0101 0010 1011 0101 1010 1101 0111 1011 1101 1111 
192  18,446,744,069,414,584,321  1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0001 
After the table, the next 10 binary unique prime have period length 170, 234, 158, 165, 147, 129, 184, 89, 208, and 312. Besides, the bits (digits in binary) of them are 65, 73, 78, 81, 82, 84, 88, 89, 96, and 97.
Biunique primes[edit]
Biunique primes are a pairs of primes having a period length shared by no other primes. For example, in binary, the biunique primes with at least one prime less than 10000 are:
prime p 
the only other prime having the same period as p  period length 

23  89  11 
29  113  28 
37  109  36 
47  178481  23 
59  3033169  58 
61  1321  60 
67  20857  66 
71  122921  35 
79  121369  39 
83  8831418697  82 
89  23  11 
97  673  48 
107  28059810762433  106 
109  37  36 
113  29  28 
139  168749965921  138 
167  57912614113275649087721  83 
193  22253377  96 
223  616318177  37 
251  4051  50 
263  10350794431055162386718619237468234569  131 
281  86171  70 
283  165768537521  94 
353  2931542417  88 
397  2113  44 
433  38737  72 
463  4982397651178256151338302204762057  231 
571  160465489  114 
577  487824887233  144 
601  1801  25 
607  1512768222413735255864403005264105839324374778520631853993  303 
631  23311  45 
641  6700417  64 
643  84115747449047881488635567801  214 
673  97  48 
727  1786393878363164227858270210279  121 
751  2139731020464054092520609592459940706818275139793055476751  375 
769  442499826945303593556473164314770689  384 
919  75582488424179347083438319  153 
1039  19709014643115560219397264671577125505264032974428376489237001990435774189483906244488746953221813209  519 
1291  83861817871925183739792206470703862766563053456867813459969184678546547694793573468589875745315081  1290 
1321  61  60 
1327  2365454398418399772605086209214363458552839866247069233  221 
1429  14449  84 
1471  252359902034571016856214298851708529738525821631  245 
1543  4965395030068548134274243124972075225434447114375481299036593442726326832727934403424309955102162841656341524725641213163998408700663382552888660520657  771 
1697  99335205800663868215396640964567095667094665346141013294320587365443384719802857319737050495099341955640963272958071602273  848 
1753  1795918038741070627  146 
1777  25781083  74 
1801  601  25 
2113  397  44 
2281  3011347479614249131  190 
2801  1114513219367157067542813609361306957257890531134775327875067038594481393220804051366788787128409731513666376851495151281817670381468528387601  1400 
2971  48912491  110 
3011  631215008947706187342830494125660733360092019659681922883823392015121754384870744044074337887482936870852519582960673945561810148710850934449712549090934572292098088972061029650939105592263256293676274598529593937386833315889748213948490958132757432166701901197169972066727635929332437543971934775961  3010 
3259  960843850986532976532466235773483492840618819232206145010143480044702708779967241439519037158800917230289  1086 
3361  88959882481  168 
3967  3296810823331827444014404831943558588631803435050404237042485765714486337505843011741487225539321479275976317423474114853376321380782906502106758766783934866952124117240484839332668914566806988602931402117416523955329423560856334826333176954575294550104263404414368761262079586842542586869780254842277261781328657636993064897732127711363870426953852536828242291991249685206783121190349820804553  1983 
4051  251  50 
4129  33770734168253651800370989375796994825389296318018601048482005531172856260013942500368975908606689  688 
4177  9857737155463  87 
4523  106788290443848295284382097033  266 
4561  51049903050598156013062477654241640657829025002976204451060261008689478158715729745160924860467530309657376827104233308157772350164622158651187694109112727796663977157921  2280 
4871  82033219963138371097689272308258116841679442057301643873942124991182012434598644913857356023840478815121709542915222280972560231358838127531337  487 
5153  54410972897  112 
5281  860573414369008969457638101533827364704684164286188824383996871471626764468219429432850649798234488791036977295952185305281368586922360240161830570497885830241813162641447900350702795124321  2640 
5347  242099935645987  198 
5431  9496792988973395279834809661569251320544014305629535339041176003993804676485199470146061766714674245857484692035146769896735263094609746643655454990307740324793870789211183639302657602006206727998357155351588374904926882678074633252877772442134938712719216422825919305415646969838444956229559805502059906228691551284152908049964563516405524127028294884716508273187514617905113012642316992618568629664046915514871575875088457784721  2715 
5881  23618256244840618857212522155851714598259422753496906641681177748710460515038403366198473773770441  1470 
6043  4475130366518102084427698737  318 
6659  67348091890626757137914773048080151982788009808953349522971703676209072447919253736713943719839321930921001920443146832964494535806286153336758808764827052922284527987735369082658226155752758837734585788583920437031679967832798697874531506375848295306002069974292647012532975382657869395896549536084026643086849168707638035984652951756449232381922841508279043449553683642165895518852273616853752192626547082225232322662203349617421624450106130361133033050996977517456780759336980017504035553443204836636663583950658161718264375035034007950563418777684540446570742277682688819816930529249402141227674467861784042184664703273065772145626307008333021727910295689307897812592176340797189662547498298629041419687123412984210580325376481846316396566413701109848755348878790562296275295266190788880122151835567771665815565611863261470167857288685014242115505182159685753576612239477286620238583071292970734389580521730540789853959607322402465845627773409459421340250476125665859926003121138412497735360569  6658 
6719  215006106257113223254503015023149432126193150293791416185445173578281597218315377296589584591228602041183907532584815068471747291177386898925622477208530115714962355294842135137890474394949339249259335407710018584480055157825387089416912233252714054247018216597994795059161567922302450277281351135838393171424038832688432240078361264161523904355539085927738753968157018550258476163852090826756157915705283413226000816151712543838581066281600650278690534719371112997393190721068136840596790525950480851370510277560248341182341805553054000587378384785994695875587394905226703149605689830257768229987714770949192483302583569799141079867597051190134078718011730508482542567284418838119000563443985593691221203060137047648713095502877775290062907208508727269017130916691676838817452529964938878349395785642430571852241837461604136374448443730175081889502056290512497717177577492736555784081731998565765598518104822516520340301701034123926767472784665776779480628821628279687651736198541330802238405154786248043073  3359 
7487  26828803997912886929710867041891989490486893845712448833  197 
8929  197107422273014301919781414466039325387889623676342705850752210599969  496 
8969  10508537584872980049787749414505440238543661684506416445249892188329191267897669657242625405655025902294996965713681247700894953567276596965114308183649957469931262029470372188492494505614207827774171575432114297123003373257035070542940532411186322417809411123684246738342720455933424175399671044286557638075591  1121 
9547  1621441292160739312484402643488810210953460916758334047593952342310982348899125523375207637304333778211869062392988099059802528019593682234941755422758885656395068722385037980657466257618112582188770921312100125511337836412531718154395821529210922389443733616354268820219577863577759459082447218927273695668223251258943006743614909639761127161704816862626236353032622115795192245125083091261029053988053316433377173895018793740052548266015018756763150731725385456332232982433576015547722563978072554378000015707071821371450842910648052930276764535303424167478747579771592484270800978561959411183367133498969236434846108865206764889977963554070295936092795484663326925724277620386077381551473009733178990983013487085676185144378484902884955972104873606558173086267499547566081780818064857671567480196001693236835136811036110768546793929610732909274227296407079545788520811837495181586420117807667033593394473  9546 
Although there are 1228 odd primes below 10000, only 21 of them are unique and 76 of them are biunique in binary.
A classic example of binary biunique primes are
 46817226351072265620777670675006972301618979214252832875068976303839400413682313921168154465151768472420980044715745858522803980473207943564433 (143 digits)
and
 527739642811233917558838216073534609312522896254707972010583175760467054896492872702786549764052643493511382273226052631979775533936351462037464331880467187717179256707148303247 (177 digits)
they are the two prime factors of the Mersenne number 2^{1061}−1. ^{[5]} Thus, the period
length of them is 1061.
As of October 2016, the largest known probable binary biunique prime is , ^{[6]} it has a period
length of 5240707 shares with only the prime 75392810903.
Similarly, we can define "triunique primes" as a triple of primes having a period
length shared by no other primes. The first few triunique primes are:
prime p  the only two other primes having the same period as p  period length 

53  157, 1613  52 
101  8101, 268501  100 
103  2143, 11119  51 
131  409891, 7623851  130 
137  953, 26317  68 
157  53, 1613  52 
163  135433, 272010961  162 
179  62020897, 18584774046020617  178 
181  54001, 29247661  180 
191  420778751, 30327152671  95 
197  19707683773, 4981857697937  196 
199  153649, 33057806959  99 
211  664441, 1564921  210 
229  457, 525313  76 
233  1103, 2089  29 
271  348031, 49971617830801  135 
307  2857, 6529  102 
317  381364611866507317969, 604462909806215075725313  316 
359  1433, 1489459109360039866456940197095433721664951999121  179 
367  55633, 37201708625305146303973352041  183 
373  951088215727633, 4611545283086450689  372 
419  3410623284654639440707, 1607792018780394024095514317003  418 
421  146919792181, 1041815865690181  420 
431  9719, 2099863  43 
439  2298041, 9361973132609  73 
443  4714692062809, 4507513575406446515845401458366741487526913  442 
457  229, 525313  76 
467  27961, 352369374013660139472574531568890678155040563007620742839120913  466 
491  15162868758218274451, 50647282035796125885000330641  490 
In binary, the smallest nunique prime are
 3, 23, 53, 149, 269, 461, 619, 389, ...
In binary, the period length of odd primes are: (sequence A014664 in the OEIS)
prime  period length 
prime  period length 
prime  period length 
prime  period length 
prime  period length 
prime  period length 
prime  period length 

3  2  79  39  181  180  293  292  421  420  557  556  673  48 
5  4  83  82  191  95  307  102  431  43  563  562  677  676 
7  3  89  11  193  96  311  155  433  72  569  284  683  22 
11  10  97  48  197  196  313  156  439  73  571  114  691  230 
13  12  101  100  199  99  317  316  443  442  577  144  701  700 
17  8  103  51  211  210  331  30  449  224  587  586  709  708 
19  18  107  106  223  37  337  21  457  76  593  148  719  359 
23  11  109  36  227  226  347  346  461  460  599  299  727  121 
29  28  113  28  229  76  349  348  463  231  601  25  733  244 
31  5  127  7  233  29  353  88  467  466  607  303  739  246 
37  36  131  130  239  119  359  179  479  239  613  612  743  371 
41  20  137  68  241  24  367  183  487  243  617  154  751  375 
43  14  139  138  251  50  373  372  491  490  619  618  757  756 
47  23  149  148  257  16  379  378  499  166  631  45  761  380 
53  52  151  15  263  131  383  191  503  251  641  64  769  384 
59  58  157  52  269  268  389  388  509  508  643  214  773  772 
61  60  163  162  271  135  397  44  521  260  647  323  787  786 
67  66  167  83  277  92  401  200  523  522  653  652  797  796 
71  35  173  172  281  70  409  204  541  540  659  658  809  404 
73  9  179  178  283  94  419  418  547  546  661  660  811  270 
In binary, the primes with given period length are: (sequence A108974 in the OEIS)
period length 
prime(s)  period length 
prime(s)  period length 
prime(s)  period length 
prime(s) 

1  (none)  26  2731  51  103, 2143, 11119  76  229, 457, 525313 
2  3  27  262657  52  53, 157, 1613  77  581283643249112959 
3  7  28  29, 113  53  6361, 69431, 20394401  78  22366891 
4  5  29  233, 1103, 2089  54  87211  79  2687, 202029703, 1113491139767 
5  31  30  331  55  881, 3191, 201961  80  4278255361 
6  (none)  31  2147483647  56  15790321  81  2593, 71119, 97685839 
7  127  32  65537  57  32377, 1212847  82  83, 8831418697 
8  17  33  599479  58  59, 3033169  83  167, 57912614113275649087721 
9  73  34  43691  59  179951, 3203431780337  84  1429, 14449 
10  11  35  71, 122921  60  61, 1321  85  9520972806333758431 
11  23, 89  36  37, 109  61  2305843009213693951  86  2932031007403 
12  13  37  223, 616318177  62  715827883  87  4177, 9857737155463 
13  8191  38  174763  63  92737, 649657  88  353, 2931542417 
14  43  39  79, 121369  64  641, 6700417  89  618970019642690137449562111 
15  151  40  61681  65  145295143558111  90  18837001 
16  257  41  13367, 164511353  66  67, 20857  91  911, 112901153, 23140471537 
17  131071  42  5419  67  193707721, 761838257287  92  277, 1013, 1657, 30269 
18  19  43  431, 9719, 2099863  68  137, 953, 26317  93  658812288653553079 
19  524287  44  397, 2113  69  10052678938039  94  283, 165768537521 
20  41  45  631, 23311  70  281, 86171  95  191, 420778751, 30327152671 
21  337  46  2796203  71  228479, 48544121, 212885833  96  193, 22253377 
22  683  47  2351, 4513, 13264529  72  433, 38737  97  11447, 13842607235828485645766393 
23  47, 178481  48  97, 673  73  439, 2298041, 9361973132609  98  4363953127297 
24  241  49  4432676798593  74  1777, 25781083  99  199, 153649, 33057806959 
25  601, 1801  50  251, 4051  75  100801, 10567201  100  101, 8101, 268501 
Period lengths[edit]
Table of period lengths from 1 to 100 (unique primes are bold)  


Unique prime in various bases[edit]
base  unique period length 

2  2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 342, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567, ... 
3  1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 24, 26, 32, 33, 36, 40, 46, 60, 63, 64, 70, 71, 72, 86, 103, 108, 128, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, ... 
4  1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, ... 
5  1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 18, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 171, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536, ... 
6  1, 2, 3, 4, 5, 6, 7, 8, 18, 21, 22, 24, 29, 30, 42, 50, 62, 71, 86, 90, 94, 118, 124, 127, 129, 144, 154, 186, 192, 214, 271, 354, 360, 411, 480, 509, 558, 575, ... 
7  3, 4, 5, 6, 8, 13, 18, 21, 28, 30, 34, 36, 46, 48, 50, 54, 55, 58, 63, 76, 84, 94, 105, 122, 131, 148, 149, 224, 280, 288, 296, 332, 352, 456, 528, 531, ... 
8  1, 2, 3, 6, 9, 18, 30, 42, 78, 87, 114, 138, 189, 303, 318, 330, 408, 462, 504, 561, ... 
9  1, 2, 4, 6, 10, 12, 16, 18, 20, 30, 32, 36, 54, 64, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540, ... 
10  1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, ... 
11  2, 4, 5, 6, 8, 9, 10, 14, 15, 17, 18, 19, 20, 27, 36, 42, 45, 52, 60, 73, 91, 104, 139, 205, 234, 246, 318, 358, 388, 403, 458, 552, ... 
12  1, 2, 3, 5, 10, 12, 19, 20, 21, 22, 56, 60, 63, 70, 80, 84, 92, 97, 109, 111, 123, 164, 189, 218, 276, 317, 353, 364, 386, 405, 456, 511, ... 
13  2, 3, 5, 6, 7, 8, 9, 12, 16, 22, 24, 28, 33, 34, 38, 78, 80, 102, 137, 140, 147, 224, 230, 283, 304, 341, 360, 372, 384, 418, 420, 436, 483, 568, 570, ... 
14  1, 3, 4, 6, 7, 14, 19, 24, 31, 33, 35, 36, 41, 55, 60, 106, 114, 129, 152, 153, 172, 222, 265, 286, 400, 448, 560, ... 
15  3, 4, 6, 7, 14, 24, 43, 54, 58, 73, 85, 93, 102, 184, 220, 221, 228, 232, 247, 291, 305, 486, 487, 505, 551, 552, ... 
16  2, 4, 6, 8, 10, 14, 20, 30, 46, 48, 52, 70, 74, 78, 150, 178, 204, 298, 306, 346, 366, 378, 400, 476, 498, 502, ... 
17  1, 2, 3, 5, 7, 8, 11, 12, 14, 15, 34, 42, 46, 47, 48, 50, 71, 77, 94, 110, 114, 147, 154, 176, 228, 235, 258, 275, 338, 350, 419, 450, 480, 515, ... 
18  1, 2, 3, 6, 14, 17, 21, 24, 30, 33, 38, 45, 46, 72, 78, 114, 146, 168, 288, 414, 440, 448, ... 
19  2, 3, 4, 6, 19, 20, 31, 34, 47, 56, 59, 61, 70, 74, 91, 92, 96, 98, 107, 120, 145, 156, 168, 242, 276, 314, 326, 337, 387, 565, ... 
20  1, 3, 4, 6, 8, 9, 10, 11, 17, 30, 98, 100, 110, 126, 154, 158, 160, 168, 178, 182, 228, 266, 270, 280, 340, 416, 480, 574, ... 
21  2, 3, 5, 6, 8, 9, 10, 11, 14, 17, 26, 43, 64, 74, 81, 104, 192, 271, 321, 335, 348, 404, 437, 445, 516, ... 
22  2, 5, 6, 7, 10, 21, 25, 26, 69, 79, 86, 93, 100, 101, 154, 158, 161, 171, 202, 214, 294, 354, 359, 424, 454, ... 
23  2, 5, 8, 11, 15, 22, 26, 39, 42, 45, 54, 56, 132, 134, 145, 147, 196, 212, 218, 252, 343, ... 
24  1, 2, 3, 4, 5, 8, 14, 19, 22, 38, 45, 53, 54, 70, 71, 117, 140, 144, 169, 186, 192, 195, 196, 430, ... 
Bibliography[edit]
 Chris K. Caldwell, Harvey Dubner, "Uniqueperiod primes", Journal of Recreational Mathematics 29:1:4348 (1998) preprint
References[edit]
 ^ Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014.
 ^ PRP Records: Probable Primes Top 10000
 ^ The Top Twenty Unique; Chris Caldwell
 ^ PRP records
 ^ The Cunningham Project
 ^ PRP records
 Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314. Zbl 0445.10009.