Unique prime
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.
No. of known terms | 102 |
---|---|
Conjectured no. of terms | Infinite |
First terms | 3, 11, 37, 101 |
Largest known term | (10270343-1)/9 |
OEIS index |
|
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 |
---|
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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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
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 lists 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)
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 1128 digits. The record has been improved many times since then. As of 2017 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).
As of September 2019, the largest known base 2 unique prime is 282589933-1, 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.
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:
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 | 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 bi-unique in binary.
A classic example of binary bi-unique primes are
- 46817226351072265620777670675006972301618979214252832875068976303839400413682313921168154465151768472420980044715745858522803980473207943564433 (143 digits)
and
- 527739642811233917558838216073534609312522896254707972010583175760467054896492872702786549764052643493511382273226052631979775533936351462037464331880467187717179256707148303247 (177 digits)
they are the two prime factors 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 |
Period lengths
Table of period lengths from 1 to 100 (unique primes are bold) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
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, ... |
Bibliography
- Chris K. Caldwell, Harvey Dubner, "Unique-period primes", Journal of Recreational Mathematics 29:1:43-48 (1998) preprint
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.