Unique prime number

In entertainment mathematics , a unique prime number or unique periodic prime number (from unique prime or unique period prime ) is a prime number for which the following applies: ${\displaystyle p\in \mathbb {P} }$

• The expansion of the decimal fraction of (i.e. the reciprocal of ) has a unique period length , that is, there is no other prime number for which has the same period length. One says "the prime number has a period of length ". ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle p}$ ${\displaystyle n\geq 1}$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$ ${\displaystyle p}$ ${\displaystyle n}$

Unique prime numbers were first studied by Samuel Yates in 1980 .

Examples

• The reciprocal of the prime number is the fraction whose decimal fraction expansion is. The period length of is thus . Of course there are also other periodic decimal fraction expansions with a period length , for example , but for is not a prime number. Also has the period length , but this fraction does not have the form , it has . There is no other fractional number of the form that the period length has. Thus is a unique prime number. ${\displaystyle p=3}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle {\frac {1}{3}}=0{,}3333333\ldots =0,{\overline {3}}}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle 1}$ ${\displaystyle n=1}$ ${\displaystyle 0{,}{\overline {1}}=0{,}1111111\ldots ={\frac {1}{9}}}$ ${\displaystyle {\frac {1}{9}}={\frac {1}{k}}}$ ${\displaystyle k=9\not \in \mathbb {P} }$ ${\displaystyle 0{,}{\overline {6}}=0{,}666666\ldots ={\frac {2}{3}}}$ ${\displaystyle n=1}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle {\frac {2}{p}}}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle 1}$ ${\displaystyle p=3}$
• The reciprocal of the prime number is the fraction whose decimal fraction expansion is. The period length of is thus . All other fractions with a period length have the form , but this fraction can at best be shortened by , by , by , by or by and get the denominator or . The only prime denominator is (because the fraction with has the period length ). So there is no other fractional number of the form which the period length has. Thus is a unique prime number. ${\displaystyle p=11}$ ${\displaystyle {\frac {1}{11}}}$ ${\displaystyle {\frac {1}{11}}=0{,}090909\ldots =0{,}{\overline {09}}}$ ${\displaystyle {\frac {1}{11}}}$ ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 0{,}{\overline {xy}}=0{,}xyxyxyxy\ldots ={\frac {xy}{99}}}$ ${\displaystyle 3}$ ${\displaystyle 9}$ ${\displaystyle 11}$ ${\displaystyle 33}$ ${\displaystyle 99}$ ${\displaystyle 33,11,9,3}$ ${\displaystyle 1}$ ${\displaystyle p=11}$ ${\displaystyle p=3}$ ${\displaystyle 1}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle n=2}$ ${\displaystyle p=11}$
• The reciprocal of the prime number is the fraction whose decimal fraction expansion is. The period length of is thus . However, the prime number also has the fraction with a period length as a reciprocal value . Thus, neither the prime nor the prime is a unique prime. ${\displaystyle p=41}$ ${\displaystyle {\frac {1}{41}}}$ ${\displaystyle {\frac {1}{41}}=0{,}0243902439\ldots =0{,}{\overline {02439}}}$ ${\displaystyle {\frac {1}{41}}}$ ${\displaystyle n=5}$ ${\displaystyle p=271}$ ${\displaystyle {\frac {1}{271}}=0{,}0036900369\ldots =0{,}{\overline {00369}}}$ ${\displaystyle n=5}$ ${\displaystyle p=41}$ ${\displaystyle p=271}$
• The smallest unique prime numbers are the following:

3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999,999,000,001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991, 909090909090909090909090909091 ... (sequence A040017 in OEIS )

The associated period lengths are as follows:

1, 2, 3, 4, 10, 12, 9, 14, 24, 36, 48, 38, 19, 23, 39, 62, ... (episode A051627 in OEIS )

Example:

In the two lists above, the two numbers and can be found in the 10th position . Thus the fraction has the period length and there is no other fraction of the form with that has the period length . ${\displaystyle 999999000001}$ ${\displaystyle 36}$ ${\displaystyle {\frac {1}{p}}={\frac {1}{999999000001}}}$ ${\displaystyle 36}$ ${\displaystyle {\frac {1}{q}}}$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle 36}$

• The 24th unique prime number has 128 digits and the corresponding fraction has a period length of 320. The prime number is: ${\displaystyle p}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle p}$

${\displaystyle p=999\ldots 999000\ldots 000999\ldots 999000\ldots 0001}$

This number starts with 32 nines, followed by 32 zeros, followed by 32 nines and 32 zeros, and it ends with one . You also write briefly . ${\displaystyle 1}$ ${\displaystyle p=(9_{32}0_{32})_{2}+1)}$

• There are currently more than 50 unique prime numbers (or unique PRP numbers , i.e. numbers that are very likely to be prime numbers, but which are currently too large to be absolutely certain) known. There are only 18 unique prime numbers that are less than and 23 unique prime numbers that are less than . ${\displaystyle 10^{50}}$ ${\displaystyle 10^{100}}$
• The currently largest probable unique prime number (as of November 22, 2018) is the following:

${\displaystyle p={\frac {10^{270343-1}}{9}}}$

It has jobs, is a repunit and was discovered in July 2007 by Maksym Voznyy and Anton Budnyy. However, this number is a PRP number , that is, it has not yet been established whether it is really prime or not because it is so large. But it fulfills many requirements for a prime number. ${\displaystyle 270343}$

• The currently largest proven unique prime (as of November 22, 2018) is the following:

${\displaystyle p=\Phi _{47498}(10)}$

She has jobs and was discovered by Ray Chandler on April 26, 2014. It can also be represented as (see the full number in the discussion section of this article ). Here is the nth circle division polynomial . ${\displaystyle 20160}$ ${\displaystyle p=\Phi _{23749}(-10)}$ ${\displaystyle \Phi _{n}(x)}$

• Following is a table that you can see which period lengths to which fractions with include. Unique prime numbers are written in yellow cells: ${\displaystyle n\leq 100}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle p\in \mathbb {P} }$

properties

• Every prime repunit ( i.e. prime numbers of the form with ones) is a unique prime number. ${\displaystyle R_{n}}$ ${\displaystyle 111\ldots 111}$ ${\displaystyle n}$

Example:

The following list gives the currently known prime repunits : ${\displaystyle n}$ ${\displaystyle R_{n}}$

2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343 (sequence A004023 in OEIS )

This includes the last four repunits and PRP numbers , so it is not yet certain whether they are really prime numbers. ${\displaystyle R_{49081},R_{86453},R_{109297}}$ ${\displaystyle R_{270343}}$

• The following two statements are equivalent:

• The prime number is a unique prime number with a period . ${\displaystyle p}$ ${\displaystyle n}$
• ${\displaystyle p^{\alpha }={\frac {\Phi _{n}(10)}{\operatorname {ggT} (\Phi _{n}(10),n)}}}$is a power of , where the nth is a pitch polynomial . ${\displaystyle p}$ ${\displaystyle \Phi _{n}(x)}$

Special case:

If a prime number, the following applies to the polynomial polynomial : ${\displaystyle n\in \mathbb {P} }$ ${\displaystyle \Phi _{n}(x)}$

${\displaystyle \Phi _{n}(x)=1+x+x^{2}+x^{3}+\ldots +x^{n-1}=\sum _{i=0}^{n-1}x^{i}}$ and thus is ${\displaystyle \Phi _{n}(10)=1+10+10^{2}+10^{3}+\ldots +10^{n-1}=\sum _{i=0}^{n-1}10^{i}}$

So for the upper sentence:

${\displaystyle p^{\alpha }={\frac {1+10+10^{2}+10^{3}+\ldots +10^{n-1}}{\operatorname {ggT} (1+10+10^{2}+10^{3}+\ldots +10^{n-1},n)}}={\frac {111\ldots 111}{\operatorname {ggT} (111\ldots 111,n)}}={\frac {R_{n}}{\operatorname {ggT} (R_{n},n)}}}$, where the -th is repunit ${\displaystyle R_{n}}$ ${\displaystyle n}$

Example:

Let be the period length . Then is . ${\displaystyle n=3\in \mathbb {P} }$ ${\displaystyle p^{\alpha }={\frac {111}{\operatorname {ggT} (111,3)}}={\frac {111}{3}}=37}$

The above list of unique prime numbers shows that for the period length is actually . ${\displaystyle p=37}$ ${\displaystyle 3}$

Normal case:

If there is no prime number, then the following applies to the polynomial division : ${\displaystyle n\not \in \mathbb {P} }$ ${\displaystyle \Phi _{n}(x)}$

${\displaystyle \Phi _{n}(x)=\!\!\!\!\prod _{1\leq k\leq n \atop \operatorname {ggT} (k,n)=1}\!\!\!\!\left(x-e^{2\pi \cdot \mathrm {i} k/n}\right)}$

Example 1:

Let be the period length . Then it is and it applies: ${\displaystyle n=9\not \in \mathbb {P} }$ ${\displaystyle \Phi _{9}(x)=x^{6}+x^{3}+1}$

${\displaystyle p^{\alpha }={\frac {\Phi _{9}(10)}{\operatorname {ggT} (\Phi _{9}(10),9)}}={\frac {10^{6}+10^{3}+1}{\operatorname {ggT} (10^{6}+10^{3}+1,9)}}={\frac {1001001}{\operatorname {ggT} (1001001,9)}}={\frac {1001001}{3}}=333667}$.

The above list of unique prime numbers shows that for the period length is actually . ${\displaystyle p=333667}$ ${\displaystyle 9}$

Example 2:

Let be the period length . Then it is and it applies: ${\displaystyle n=1\not \in \mathbb {P} }$ ${\displaystyle \Phi _{1}(x)=x-1}$

${\displaystyle p^{\alpha }={\frac {\Phi _{1}(10)}{\operatorname {ggT} (\Phi _{1}(10),1)}}={\frac {10-1}{\operatorname {ggT} (10-1,9)}}={\frac {9}{\operatorname {ggT} (9,1)}}={\frac {9}{1}}=9=3^{2}}$.

The above list of unique prime numbers shows that for the period length is actually . ${\displaystyle p=3}$ ${\displaystyle 1}$

Example 3:

Let be the period length . Then it is and it applies: ${\displaystyle n=6\not \in \mathbb {P} }$ ${\displaystyle \Phi _{6}(x)=x^{2}-x+1}$

${\displaystyle p^{\alpha }={\frac {\Phi _{6}(10)}{\operatorname {ggT} (\Phi _{6}(10),6)}}={\frac {10^{2}-10+1}{\operatorname {ggT} (10^{2}-10+1,6)}}={\frac {91}{\operatorname {ggT} (91,6)}}={\frac {91}{1}}=91}$.

But it is not a prime number, so there is also no unique prime number with a period length . Instead, the expansions of decimal fractions and have the period length . ${\displaystyle 91=7\cdot 13\not \in P}$ ${\displaystyle 6}$ ${\displaystyle {\frac {1}{7}}}$ ${\displaystyle {\frac {1}{13}}}$ ${\displaystyle n=6}$

Unsolved problems

• It is believed that there are infinitely many unique prime numbers (this would infer from another mathematical conjecture, namely that there are infinitely many prime repunits).

Unique prime numbers in the dual system

Unique prime numbers depend on the basis used to count. In the sections above, unique prime numbers were considered as the base , i.e. in the decimal system . This section deals with unique prime numbers in the dual system , i.e. with a base . ${\displaystyle b=10}$ ${\displaystyle b=2}$

A prime number is a unique prime number to the base b = 2 , if and only if the following applies: ${\displaystyle p\in \mathbb {P} }$

• The fraction is based on the period length . There is no other prime number for which the fraction at the base also has the period length . ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle b=2}$ ${\displaystyle n\geq 1}$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$ ${\displaystyle b=2}$ ${\displaystyle n}$

Examples

• A unique prime number in the dual system is the number : ${\displaystyle p=5=4+1={\underline {1}}\cdot 2^{2}+{\underline {0}}\cdot 2^{1}+{\underline {1}}\cdot 2^{0}=101_{2}}$

It is

${\displaystyle {\begin{aligned}{\frac {1}{5}}=0,2&={\frac {0}{2}}+{\frac {0}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {0}{32}}+{\frac {0}{64}}+{\frac {1}{128}}+{\frac {1}{256}}+\ldots =0,125+0,0625+\ldots =\\&={\underline {0}}\cdot 2^{-1}+{\underline {0}}\cdot 2^{-2}+{\underline {1}}\cdot 2^{-3}+{\underline {1}}\cdot 2^{-4}+{\underline {0}}\cdot 2^{-5}+{\underline {0}}\cdot 2^{-6}+{\underline {1}}\cdot 2^{-7}+{\underline {1}}\cdot 2^{-8}+\ldots =0,00110011\ldots =0,{\overline {0011}}_{2}\end{aligned}}}$

a number with period length, periodic in the dual system . There is no other prime number whose fraction in the dual system has a period length of . Thus is a unique prime number in the dual system. ${\displaystyle n=4}$ ${\displaystyle p}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle n=4}$ ${\displaystyle p=5}$

• For the number is a number that is not periodic in the dual system (i.e. with a period length ). Although there is no other prime number whose fraction in the dual system has a period length of , there is still no unique prime number in the dual system, because there must be. ${\displaystyle p=2={\underline {1}}\cdot 2^{1}+{\underline {0}}\cdot 2^{0}=10_{2}}$ ${\displaystyle {\frac {1}{p}}={\frac {1}{2}}=0,5={\underline {1}}\cdot 2^{-1}=0,1_{2}}$ ${\displaystyle n=0}$ ${\displaystyle p}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle n=0}$ ${\displaystyle p=2}$ ${\displaystyle n\geq 1}$
• The smallest unique prime numbers in the dual system are the following, each written in the decimal system:

3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, 2731, 5419, 8191, 43691, 61681, 65537, 87211, 131071, 174763, 262657, 524287, 599479, 2796203, 15790321, 18837001, 22366891, 715827883, 2147483647, 4278255361, ... ( continuation A144755 in OEIS )

The associated period lengths are as follows:

2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, 26, 42, 13, 34, 40, 32, 54, 17, 38, 27, 19, 33, 46, 56, 90, 78, 62, 31, 80, 120, 126, 150, 86, 98, 49, 69, 65, 174, 77, 93, 122, 61, 85, 192, 170, 234, 158, 165, 147, 129, 184, 89, 208, 312, ... (follow A247071 in OEIS )

If you want to order the unique prime numbers in the dual system according to their period length , you get the sequence A161509 in OEIS . The sorted list of the associated period lengths is then the sequence A161508 in OEIS . ${\displaystyle n}$ ${\displaystyle n}$

• The currently (as of December 23, 2018) largest known unique prime number in the dual system is the following:

${\displaystyle p=2^{82589933}-1}$

She has jobs and was discovered by Patrick Laroche on December 21, 2018. It is also the largest known prime number and therefore also the largest known Mersenne prime number . The corresponding fraction , written in the dual system, has the period length and there is not a single other prime number whose fraction has the same period length. ${\displaystyle 24.862.048}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle n=82589933}$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$

• The currently (as of July 21, 2018) largest known unique (but not yet finally proven) prime number in the dual system, which is not also Mersenne prime number, is the following:

${\displaystyle p={\frac {2^{13372531}+1}{3}}}$

She has jobs and was discovered by Ryan Propper in September 2013. However, it is still too large to be able to say with certainty that it is a prime number. It fulfills many prime number properties and is a PRP number . If its primality is proven, it is a Wagstaff prime number . The corresponding fraction , written in the dual system, has the period length and there is not a single other prime number whose fraction has the same period length. ${\displaystyle 4.025.533}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle n=2\cdot 13372531=26745062}$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$

• The currently (as of July 21, 2018) largest known unique (and also proven) prime number in the dual system, which is not simultaneously a Mersenne prime number, is the following:

${\displaystyle p={\frac {2^{83339}+1}{3}}}$

She has jobs and was discovered by Tom Wu on September 17th, 2014. It is currently the largest known Wagstaff prime. The corresponding fraction has, written in the dual system, the period length . ${\displaystyle 25088}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle n=2\cdot 83339=166678}$

• The currently (as of July 21, 2018) largest known unique prime number in the dual system, which is neither Mersenne prime nor Wagstaff prime (but unfortunately a PRP number ), is the following:

${\displaystyle p={\frac {2^{4101572}+1}{17}}={\frac {16^{1025393}+1}{17}}}$

She has jobs and was discovered by Paul Bourdelais in August 2014. ${\displaystyle 1.234.695}$

properties

• Every Fermat prime number is a unique prime number in the dual system. Their period length is a power of two with . ${\displaystyle F_{n}=2^{2^{n}}+1}$ ${\displaystyle 2^{n}}$ ${\displaystyle n\in \mathbb {N} ,n>0}$
• Every Mersenne prime is a unique prime in the dual system. Their period length is a prime number . ${\displaystyle M_{n}=2^{n}-1}$ ${\displaystyle n\in \mathbb {P} }$
• Every Wagstaff prime number is a unique prime number in the dual system. Their period length is twice an odd prime number with . ${\displaystyle p={\frac {2^{q}+1}{3}}}$ ${\displaystyle n=2\cdot q}$ ${\displaystyle q\in \mathbb {P} ,q\geq 3}$
• Let and be a natural number . Then: ${\displaystyle n\not =1}$ ${\displaystyle n\not =6}$

There is at least one prime number , which in the dual system has the period length . ${\displaystyle p}$ ${\displaystyle n}$

Proof: This statement is valid because of Zsigmondy's theorem ( en )

• Let be a natural number with ( so have the form with ). Then: ${\displaystyle n>20}$ ${\displaystyle n\equiv 4{\pmod {8}}}$ ${\displaystyle n}$ ${\displaystyle n=8k+4}$ ${\displaystyle k\in \mathbb {N} }$

There are at least two prime numbers that have the period length in the dual system . ${\displaystyle p_{1},p_{2}}$ ${\displaystyle n}$

Thus there is never a unique prime number for the base . ${\displaystyle n\equiv 4{\pmod {8}}}$ ${\displaystyle b=2}$

Proof: This statement is true because of the factorization of Aurifeuille ( s )

• The following two statements are equivalent:

• The prime number is a unique prime number in the dual system with a period . ${\displaystyle p}$ ${\displaystyle n}$
• ${\displaystyle p^{\alpha }={\frac {\Phi _{n}(2)}{\operatorname {ggT} (\Phi _{n}(2),n)}}}$is a power of with , where the nth is a pitch polynomial . ${\displaystyle p}$ ${\displaystyle \alpha \in \mathbb {N} }$ ${\displaystyle \Phi _{n}(x)}$

Example:

The only known ones for which the above numerator is composed, but the above total expression is prime, are the following: ${\displaystyle n}$ ${\displaystyle \Phi _{n}(2)}$ ${\displaystyle {\frac {\Phi _{n}(2)}{\operatorname {ggT} (\Phi _{n}(2),n)}}}$

18, 20, 21, 54, 147, 342, 602, 889

In these cases it obviously has a factor that is also a factor of . ${\displaystyle \Phi _{n}(2)}$ ${\displaystyle n}$

All other known unique prime numbers for the base have the form . ${\displaystyle b=2}$ ${\displaystyle \Phi _{n}(2)}$

No prime number is known for which is in the above formula . The following applies to all known unique prime numbers in the dual system . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle \alpha >1}$ ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle \alpha =1}$

Unsolved problems

• It is conjectured that there are infinitely many unique primes for the base (this would infer from another mathematical conjecture, namely that there are infinitely many Mersenne primes). ${\displaystyle b=2}$
• It is assumed that there are no Wieferich prime numbers that are also unique prime numbers in the dual system.

Unique prime numbers in other number systems

A prime number is a b unique prime number to the base , if and only if the following applies: ${\displaystyle p\in \mathbb {P} }$

• The fraction is based on the period length . There is no other prime number for which the fraction at the base also has the period length . ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle b}$ ${\displaystyle n\geq 1}$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$ ${\displaystyle b}$ ${\displaystyle n}$

properties

• The following three statements are equivalent:

• ${\displaystyle p}$is a unique prime number as a base (the fraction has the period length as a base ). ${\displaystyle b}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle b}$ ${\displaystyle n}$
• ${\displaystyle p}$is the only prime divisor of the nth circle division polynomial which does not divide the period length . ${\displaystyle \Phi _{n}(b)}$ ${\displaystyle n}$
• Case 1: is even: ${\displaystyle b}$

${\displaystyle R_{n}(b)={\frac {\Phi _{n}(b)}{\operatorname {ggT} (\Phi _{n}(b),n)}}=p^{\alpha }}$is a power of with ${\displaystyle p}$ ${\displaystyle \alpha \in \mathbb {N} }$

Case 2: is odd: ${\displaystyle b}$

${\displaystyle R_{n}(b)={\frac {\Phi _{n}(b)}{\operatorname {ggT} (\Phi _{n}(b),n)}}=2^{\beta }\cdot p^{\alpha }}$is a power of two times the power of with ${\displaystyle p}$ ${\displaystyle \alpha ,\beta \in \mathbb {N} ,\alpha >0,\beta \geq 0}$

Unique prime numbers in the decimal system or in the dual system therefore fall into case 1.

• Let the prime number divide the base . Then: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle b}$

• The prime number is not a unique base prime . ${\displaystyle p}$ ${\displaystyle b}$
• The base of the fraction has the period length , i.e. it has no period. ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle b}$ ${\displaystyle n=0}$

Proof of the first claim:

If the base is a divisor , it is also a divisor of and therefore not a divisor of the larger number . So is too coprime . The circle division polynomial is defined in such a way that it has to divide. Thus is also and coprime and therefore it is not a factor of . So there can not be a unique prime number to base . ${\displaystyle p}$ ${\displaystyle b}$ ${\displaystyle p}$ ${\displaystyle b^{n}}$ ${\displaystyle 1}$ ${\displaystyle b^{n}+1}$ ${\displaystyle p}$ ${\displaystyle b^{n}-1}$ ${\displaystyle \Phi _{n}(b)}$ ${\displaystyle b^{n}-1}$ ${\displaystyle p}$ ${\displaystyle \Phi _{n}(b)}$ ${\displaystyle p}$ ${\displaystyle {\frac {\Phi _{n}(b)}{\operatorname {ggT} (\Phi _{n}(b),n)}}}$ ${\displaystyle p}$ ${\displaystyle b}$ ${\displaystyle \Box }$

• Be . Then: ${\displaystyle n\in \mathbb {N} }$

There is at least one prime number for which to base the period length has, with the exception of the following cases: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle b}$ ${\displaystyle n}$

• ${\displaystyle b=2}$and or ${\displaystyle n=1}$ ${\displaystyle n=6}$
• ${\displaystyle n=2}$and with ${\displaystyle b=2^{k}-1}$ ${\displaystyle k\in \mathbb {N} ,k>0}$

Proof: This statement is valid because of Zsigmondy's theorem ( en )

Examples

The following is a list of prime numbers for which the fraction has the period length given the base . Unique prime numbers are written in yellow cells: ${\displaystyle p}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle b\leq 24}$ ${\displaystyle n\leq 24}$

The following is a list of the period lengths of fractions of the form with the first 34 prime numbers for various bases . If the prime number is a divisor of the base , the decimal fraction development ends, so the period length is . If the prime number is a unique prime number as the base , the period length is written in a yellow cell: ${\displaystyle n}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle p\leq 139}$ ${\displaystyle b\leq 24}$ ${\displaystyle p}$ ${\displaystyle b}$ ${\displaystyle 0}$ ${\displaystyle p}$ ${\displaystyle b}$ ${\displaystyle n}$

A table now follows, from which you can see the smallest period lengths (up to and including ) for which the fraction has a unique length. There is therefore no other prime number on the given basis with the same period length. In addition, the associated unique prime number is also given, the fraction of which has this period length . ${\displaystyle n}$ ${\displaystyle n=600}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle b\leq 24}$ ${\displaystyle p}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle n}$

Bi-Unique Prime Numbers

The two prime numbers and are called bi-unique prime numbers (from bi-unique prime ) if: ${\displaystyle p_{1}\in \mathbb {P} }$ ${\displaystyle p_{2}\in \mathbb {P} }$

• The two fractions and have the same period length ${\displaystyle {\frac {1}{p_{1}}}}$ ${\displaystyle {\frac {1}{p_{2}}}}$ ${\displaystyle n}$
• There is no other prime number , so this period length has ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$ ${\displaystyle n}$

Examples

• Let be the base and the period length . Then applies to the circle division polynomial and for : ${\displaystyle b=10}$ ${\displaystyle n=6}$ ${\displaystyle \Phi _{n}(b)}$ ${\displaystyle R_{n}(b)}$

${\displaystyle \Phi _{n}(b)=\Phi _{6}(10)=10^{2}-10+1=91}$

${\displaystyle R_{n}(b)=R_{6}(10)={\frac {\Phi _{n}(b)}{\operatorname {ggT} (\Phi _{n}(b),n)}}={\frac {91}{\operatorname {ggT} (91,6)}}={\frac {91}{1}}=91=7\cdot 13}$

Thus, and have the same period length (in particular is and ). The two prime numbers and are therefore bi-unique prime numbers for the base . ${\displaystyle p_{1}=7}$ ${\displaystyle p_{2}=13}$ ${\displaystyle n=6}$ ${\displaystyle {\frac {1}{7}}=0,{\overline {142857}}}$ ${\displaystyle {\frac {1}{13}}=0,{\overline {076923}}}$ ${\displaystyle p_{1}=7}$ ${\displaystyle p_{2}=13}$ ${\displaystyle b=10}$

• Let be the base and the period length . Then applies to the circle division polynomial and for : ${\displaystyle b=2}$ ${\displaystyle n=11}$ ${\displaystyle \Phi _{n}(b)}$ ${\displaystyle R_{n}(b)}$

${\displaystyle \Phi _{n}(b)=\Phi _{11}(2)=2^{10}+2^{9}+2^{8}+2^{7}+2^{6}+2^{5}+2^{4}+2^{3}+2^{2}+2^{1}+1=2047}$

${\displaystyle R_{n}(b)=R_{11}(2)={\frac {\Phi _{n}(b)}{\operatorname {ggT} (\Phi _{n}(b),n)}}={\frac {2047}{\operatorname {ggT} (2047,11)}}={\frac {2047}{1}}=2047=23\cdot 89}$

Thus, and have the same period length (in particular is and ). The two prime numbers and are therefore bi-unique prime numbers for the base . ${\displaystyle p_{1}=23}$ ${\displaystyle p_{2}=89}$ ${\displaystyle n=11}$ ${\displaystyle {\frac {1}{23}}=0,{\overline {00001011001}}_{2}}$ ${\displaystyle {\frac {1}{89}}=0,{\overline {00000010111}}_{2}}$ ${\displaystyle p_{1}=23}$ ${\displaystyle p_{2}=89}$ ${\displaystyle b=2}$

• There are 1,228 odd prime numbers among 10,000, but only 21 of them are unique in the binary system and 76 of them are bi-unique.
• The two prime factors (143 digits) and (177 digits) of the Mersenne number are bi-unique prime numbers for the base with a period length . The two prime numbers are: ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle M_{1061}=2^{1061}-1}$ ${\displaystyle b=2}$ ${\displaystyle n=1061}$

${\displaystyle p_{1}=46817226351072265620777670675006972301618979214252832875068976303839400413682313921168154465151768472420980044715745858522803980473207943564433}$

${\displaystyle p_{2}=527739642811233917558838216073534609312522896254707972010583175760467054896492872702786549764052643493511382273226052631979775533936351462037464331880467187717179256707148303247}$

• The currently largest known bi-unique prime number (as of August 18, 2018) is currently still a PRP number (so only very likely a prime number due to its size) and is:

${\displaystyle p_{1}={\frac {2^{5240707}-1}{75392810903}}}$

It was discovered by Tony Perst in July 2016 and has 1,577,600 jobs. The period length is the associated prime number . ${\displaystyle n=5240707}$ ${\displaystyle p_{2}=75392810903}$

• The following two lists give the smallest bi-unique prime numbers and for the bases or for which both and have the same period length : ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle b=2}$ ${\displaystyle b=10}$ ${\displaystyle {\frac {1}{p_{1}}}}$ ${\displaystyle {\frac {1}{p_{2}}}}$ ${\displaystyle n}$

Tri-Unique Prime Numbers

Analogous to the bi-unique prime numbers, one can also define tri-unique prime numbers:

The three prime numbers are called tri-unique prime numbers (from tri-unique prime ) if: ${\displaystyle p_{1},p_{2},p_{3}\in \mathbb {P} }$

• The three fractions and have the same period length ${\displaystyle {\frac {1}{p_{1}}},{\frac {1}{p_{2}}}}$ ${\displaystyle {\frac {1}{p_{3}}}}$ ${\displaystyle n}$
• There is no other prime number , so this period length has ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$ ${\displaystyle n}$

Examples

• Let be the base and the period length . Then applies to the circle division polynomial and for : ${\displaystyle b=10}$ ${\displaystyle n=13}$ ${\displaystyle \Phi _{n}(b)}$ ${\displaystyle R_{n}(b)}$

${\displaystyle \Phi _{n}(b)=\Phi _{13}(10)=\sum _{i=0}^{12}10^{i}=1.111.111.111.111}$

${\displaystyle R_{n}(b)=R_{13}(10)={\frac {\Phi _{n}(b)}{\operatorname {ggT} (\Phi _{n}(b),n)}}={\frac {\Phi _{13}(10)}{\operatorname {ggT} (\Phi _{13}(10),13)}}={\frac {\Phi _{13}(10)}{1}}=1.111.111.111.111=53\cdot 79\cdot 265371653}$

Thus , and have the same period length . The three prime numbers , and are therefore tri-unique prime numbers for the base . ${\displaystyle p_{1}=53}$ ${\displaystyle p_{2}=79}$ ${\displaystyle p_{3}=265371653}$ ${\displaystyle n=13}$ ${\displaystyle p_{1}=53}$ ${\displaystyle p_{2}=79}$ ${\displaystyle p_{3}=265371653}$ ${\displaystyle b=10}$

• Let be the base and the period length . Then applies to the circle division polynomial and for : ${\displaystyle b=2}$ ${\displaystyle n=29}$ ${\displaystyle \Phi _{n}(b)}$ ${\displaystyle R_{n}(b)}$

${\displaystyle \Phi _{n}(b)=\Phi _{29}(2)=\sum _{i=0}^{28}2^{i}=536870911}$

${\displaystyle R_{n}(b)=R_{29}(2)={\frac {\Phi _{n}(b)}{\operatorname {ggT} (\Phi _{n}(b),n)}}={\frac {536870911}{\operatorname {ggT} (536870911,29)}}={\frac {536870911}{1}}=536870911=233\cdot 1103\cdot 2089}$

Thus , and have the same period length . The three prime numbers , and are therefore tri-unique prime numbers for the base . ${\displaystyle p_{1}=233}$ ${\displaystyle p_{2}=1103}$ ${\displaystyle p_{3}=2089}$ ${\displaystyle n=29}$ ${\displaystyle p_{1}=233}$ ${\displaystyle p_{2}=1103}$ ${\displaystyle p_{3}=2089}$ ${\displaystyle b=2}$

• The following two lists give the smallest tri-unique prime numbers and the base up to and the base up to, for which both and have the same period length : ${\displaystyle p_{1},p_{2}}$ ${\displaystyle p_{3}}$ ${\displaystyle b=2}$ ${\displaystyle p<1000}$ ${\displaystyle b=10}$ ${\displaystyle n\leq 200}$ ${\displaystyle {\frac {1}{p_{1}}},{\frac {1}{p_{2}}}}$ ${\displaystyle {\frac {1}{p_{3}}}}$ ${\displaystyle n}$

Generalization: n-Unique Prime Numbers

The prime numbers are called n-unique prime numbers (from the English n-unique prime ) if: ${\displaystyle n}$ ${\displaystyle p_{1},p_{2},\ldots p_{n}\in \mathbb {P} }$

• The fractions have the same period length . ${\displaystyle n}$ ${\displaystyle {\frac {1}{p_{1}}},{\frac {1}{p_{2}}},\ldots ,{\frac {1}{p_{n}}}}$ ${\displaystyle n}$
• There is no other prime so that it has a period length . ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle {\frac {1}{q}}}$ ${\displaystyle n}$

Examples

• The following prime numbers are the smallest n-unique prime numbers based on increasing : ${\displaystyle b=2}$ ${\displaystyle n=1,2,3,\ldots }$

3, 23, 53, 149, 269, 461, 619, 389, ...

Example:

The number is in the 6th position in the list above . This means that it is the smallest prime number that belongs to a 6-unique prime number tuple to the base . ${\displaystyle p=461}$ ${\displaystyle p_{1}=461}$ ${\displaystyle (p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})}$ ${\displaystyle b=2}$

• The following prime numbers are the smallest n-unique prime numbers based on increasing : ${\displaystyle b=10}$ ${\displaystyle n=1,2,3,\ldots }$

3, 7, 23, 47, 163, 149, ...

Example:

The number is at the 5th position in the list above . This means that it is the smallest prime number that belongs to a 5-unique prime number tuple to the base . ${\displaystyle p=163}$ ${\displaystyle p_{1}=163}$ ${\displaystyle (p_{1},p_{2},p_{3},p_{4},p_{5})}$ ${\displaystyle b=10}$

Individual evidence

1. ^ Samuel Yates: Periods of unique primes. Mathematics Magazine 53 , 1980, p. 314 , accessed July 16, 2018 . 
2. ^ Giovanni Di Maria: Known REPUNIT Primes. The Repunit Primes Project, accessed July 16, 2018 . 
3. ↑ ^{a b} Chris K. Caldwell: The Top Twenty: Unique. Prime Pages, accessed June 21, 2018 . 
4. ↑ Phi (47498,10) on Primo Top-20
5. ↑ Phi (23749, -10) on Prime Pages
6. ^ Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000, Search for: (10 ^ x-1) / 9. PRP Records, accessed July 16, 2018 . 
7. ↑ Chris K. Caldwell, Harvey Dubner: Unique-period primes. Journal of Recreational Mathematics 29 (1), 1963, pp. 475-478 , accessed July 16, 2018 . 
8. ↑ Eric W. Weisstein : Unique Prime . In: MathWorld (English).
9. ↑ Chris K. Caldwell: Repunit. Prime Pages, accessed July 16, 2018 . 
10. ↑ 2 ^82589933 -1 on Prime Pages
11. ^ Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000, Search for: (2 ^ n + 1) / 3. PRP Records, accessed July 21, 2018 . 
12. ↑ (2 ⁸³³³⁹ +1) / 3 on Prime Pages
13. ^ Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000, ranked 16. PRP Records, accessed on July 21, 2018 . 
14. ^ Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000, Search for: (2 ^ a-1) / b. PRP Records, accessed August 18, 2018 . 

Web links

• Eric W. Weisstein : Unique Prime . In: MathWorld (English).
• Chris K. Caldwell: Unique Prime. Prime Pages, accessed December 23, 2018 . 
• Chris K. Caldwell: Unique (Period) Primes and the Factorization of Cyclotomic Polynomial Minus One. (PDF) Mathematica Japonica 26 (1), 1997, pp. 189–195 , accessed on December 23, 2018 . 
• Chris K. Caldwell, Harvey Dubner: Unique-Period Primes. (PDF) Journal of Recreational Mathematics 29 (1), 1998, pp. 43–48 , accessed on December 23, 2018 . 
• R. Ondrejka, Chris K. Caldwell, Harvey Dubner: The Top Ten Prime Numbers. (PDF) 2001, pp. 1–96 (92) , accessed on December 23, 2018 .