Unique prime
Number of known terms | 102 |
---|---|
Conjectured number of terms | Infinite |
First terms | 3, 11, 37, 101 |
Largest known term | (10270343-1)/9 |
OEIS index | A040017 |
In number theory, a unique 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. 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.
Period of a prime in base b
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= 2k − 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
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 Rn(b).
Sketch of the proof of the characterization of primes of period n |
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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 = bm − 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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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The mention "terminate" means that the prime divides the base, and thus that the representation of its reciprocal is finite.
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Table of primes of a given period (up to 24) in bases up to 24 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Bold for unique primes.
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Decimal unique primes
At present, more than fifty unique primes or probable primes are known. However, there are only twenty-three unique primes below 10100. The following table gives an overview of all 23 unique primes below 10100 (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)
Period length | 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 to the reciprocal of 7 (0.142857142857142857...)
Just after the table, the twenty-fourth unique prime has 128 digits and period length 320. It can be written as (932032)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 the repunit (10270343-1)/9 is the largest known probable unique prime.[2]
In 1996 the largest proven unique prime was (101132 + 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 2014 the largest proven unique prime is , it has 20160 digits.[3]
Binary unique primes
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. 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 274207281-1, 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 proved 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.
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.
There are 52 unique primes in base 2 below 264, they are:
Period length | 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.
Bi-unique primes
Bi-unique primes are a pairs of primes having a period length shared by no other primes. For example, in binary, the bi-unique 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 | 197090146431155602193972646715771255052640329744283764892370019904357741 89483906244488746953221813209 |
519 |
1291 | 838618178719251837397922064707038627665630534568678134599691846785465476 94793573468589875745315081 |
1290 |
1321 | 61 | 60 |
1327 | 2365454398418399772605086209214363458552839866247069233 | 221 |
1429 | 14449 | 84 |
1471 | 252359902034571016856214298851708529738525821631 | 245 |
1543 | 496539503006854813427424312497207522543444711437548129903659344272632683 272793440342430995510216284165634152472564121316399840870066338255288866 0520657 |
771 |
1697 | 993352058006638682153966409645670956670946653461410132943205873654433847 19802857319737050495099341955640963272958071602273 |
848 |
1753 | 1795918038741070627 | 146 |
1777 | 25781083 | 74 |
1801 | 601 | 25 |
2113 | 397 | 44 |
2281 | 3011347479614249131 | 190 |
2801 | 111451321936715706754281360936130695725789053113477532787506703859448139 3220804051366788787128409731513666376851495151281817670381468528387601 |
1400 |
2971 | 48912491 | 110 |
3011 | 631215008947706187342830494125660733360092019659681922883823392015121754 384870744044074337887482936870852519582960673945561810148710850934449712 549090934572292098088972061029650939105592263256293676274598529593937386 833315889748213948490958132757432166701901197169972066727635929332437543 971934775961 |
3010 |
3259 | 960843850986532976532466235773483492840618819232206145010143480044702708 779967241439519037158800917230289 |
1086 |
3361 | 88959882481 | 168 |
3967 | 329681082333182744401440483194355858863180343505040423704248576571448633 750584301174148722553932147927597631742347411485337632138078290650210675 876678393486695212411724048483933266891456680698860293140211741652395532 942356085633482633317695457529455010426340441436876126207958684254258686 978025484227726178132865763699306489773212771136387042695385253682824229 1991249685206783121190349820804553 |
1983 |
4051 | 251 | 50 |
4129 | 337707341682536518003709893757969948253892963180186010484820055311728562 60013942500368975908606689 |
688 |
4177 | 9857737155463 | 87 |
4523 | 106788290443848295284382097033 | 266 |
4561 | 510499030505981560130624776542416406578290250029762044510602610086894781 587157297451609248604675303096573768271042333081577723501646221586511876 94109112727796663977157921 |
2280 |
4871 | 820332199631383710976892723082581168416794420573016438739421249911820124 34598644913857356023840478815121709542915222280972560231358838127531337 |
487 |
5153 | 54410972897 | 112 |
5281 | 860573414369008969457638101533827364704684164286188824383996871471626764 468219429432850649798234488791036977295952185305281368586922360240161830 570497885830241813162641447900350702795124321 |
2640 |
5347 | 242099935645987 | 198 |
5431 | 949679298897339527983480966156925132054401430562953533904117600399380467 648519947014606176671467424585748469203514676989673526309460974664365545 499030774032479387078921118363930265760200620672799835715535158837490492 688267807463325287777244213493871271921642282591930541564696983844495622 955980550205990622869155128415290804996456351640552412702829488471650827 3187514617905113012642316992618568629664046915514871575875088457784721 |
2715 |
5881 | 236182562448406188572125221558517145982594227534969066416811777487104605 15038403366198473773770441 |
1470 |
6043 | 4475130366518102084427698737 | 318 |
6659 | 673480918906267571379147730480801519827880098089533495229717036762090724 479192537367139437198393219309210019204431468329644945358062861533367588 087648270529222845279877353690826582261557527588377345857885839204370316 799678327986978745315063758482953060020699742926470125329753826578693958 965495360840266430868491687076380359846529517564492323819228415082790434 495536836421658955188522736168537521926265470822252323226622033496174216 244501061303611330330509969775174567807593369800175040355534432048366366 635839506581617182643750350340079505634187776845404465707422776826888198 169305292494021412276744678617840421846647032730657721456263070083330217 279102956893078978125921763407971896625474982986290414196871234129842105 803253764818463163965664137011098487553488787905622962752952661907888801 221518355677716658155656118632614701678572886850142421155051821596857535 766122394772866202385830712929707343895805217305407898539596073224024658 45627773409459421340250476125665859926003121138412497735360569 |
6658 |
6719 | 215006106257113223254503015023149432126193150293791416185445173578281597 218315377296589584591228602041183907532584815068471747291177386898925622 477208530115714962355294842135137890474394949339249259335407710018584480 055157825387089416912233252714054247018216597994795059161567922302450277 281351135838393171424038832688432240078361264161523904355539085927738753 968157018550258476163852090826756157915705283413226000816151712543838581 066281600650278690534719371112997393190721068136840596790525950480851370 510277560248341182341805553054000587378384785994695875587394905226703149 605689830257768229987714770949192483302583569799141079867597051190134078 718011730508482542567284418838119000563443985593691221203060137047648713 095502877775290062907208508727269017130916691676838817452529964938878349 395785642430571852241837461604136374448443730175081889502056290512497717 177577492736555784081731998565765598518104822516520340301701034123926767 472784665776779480628821628279687651736198541330802238405154786248043073 |
3359 |
7487 | 26828803997912886929710867041891989490486893845712448833 | 197 |
8929 | 197107422273014301919781414466039325387889623676342705850752210599969 | 496 |
8969 | 105085375848729800497877494145054402385436616845064164452498921883291912 678976696572426254056550259022949969657136812477008949535672765969651143 081836499574699312620294703721884924945056142078277741715754321142971230 033732570350705429405324111863224178094111236842467383427204559334241753 99671044286557638075591 |
1121 |
9547 | 162144129216073931248440264348881021095346091675833404759395234231098234 889912552337520763730433377821186906239298809905980252801959368223494175 542275888565639506872238503798065746625761811258218877092131210012551133 783641253171815439582152921092238944373361635426882021957786357775945908 244721892727369566822325125894300674361490963976112716170481686262623635 303262211579519224512508309126102905398805331643337717389501879374005254 826601501875676315073172538545633223298243357601554772256397807255437800 001570707182137145084291064805293027676453530342416747874757977159248427 080097856195941118336713349896923643484610886520676488997796355407029593 609279548466332692572427762038607738155147300973317899098301348708567618 514437848490288495597210487360655817308626749954756608178081806485767156 748019600169323683513681103611076854679392961073290927422729640707954578 8520811837495181586420117807667033593394473 |
9546 |
Although there are 1228 odd primes below 10000, only 21 of them are unique and 76 of them are bi-unique in binary.
A classic example of binary bi-unique primes are
- 468172263510722656207776706750069723016189792142528328750689763038394004
13682313921168154465151768472420980044715745858522803980473207943564433 (143 digits)
and
- 527739642811233917558838216073534609312522896254707972010583175760467054
896492872702786549764052643493511382273226052631979775533936351462037464
331880467187717179256707148303247 (177 digits)
they are the two prime factor of the Mersenne number 21061−1. [5] Thus, the period length of them is 1061.
As of October 2016, the largest known probable binary bi-unique prime is , [6] it has a period length of 5240707 shares with only the prime 75392810903.
Similarly, we can define "tri-unique primes" as a triple of primes having a period length shared by no other primes. The first few tri-unique 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 n-unique 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 |
Unique prime in various bases
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, ... |
References
- ↑ 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.