Long prime

In number theory , a long prime number with base b is a prime number , for which the following applies: ${\displaystyle p\in \mathbb {P} }$

• ${\displaystyle b\in \mathbb {N} }$is a natural number , so that does not divide is ${\displaystyle p}$ ${\displaystyle b}$
• ${\displaystyle x={\frac {b^{p-1}-1}{p}}}$is a cyclic number .

The expression long prime (from the English long prime , but also full reptend prime , full repetend prime or proper prime ) was first mentioned by John Horton Conway and Richard Kenneth Guy in their book The Book of Numbers .

Derivation of the definition based on examples

If you look at the base (i.e. the decimal system ), you usually don't mention it. This section is about that base . ${\displaystyle b=10}$ ${\displaystyle b=10}$

A fraction , for example, can also be written with commas: . These decimal numbers stop, as in the case , or are infinitely long, as in, for example . This infinite repetition of the same digits is called a period and is written . The period can also be longer, for example with . Most of the time the digits of the period change when you multiply the number (for example is ). But there are periodic fractions whose digits do not change after a multiplication , as for example with . To make these six decimal places into a whole number, you multiply them with , i.e. with, and you get the number . If you subtract now , you get the whole number . If you multiply this number by again , you get: ${\displaystyle n={\frac {1}{2}}}$ ${\displaystyle n={\frac {1}{2}}=0{,}5}$ ${\displaystyle n={\frac {1}{2}}=0{,}5}$ ${\displaystyle n={\frac {1}{3}}=0{,}333333\ldots }$ ${\displaystyle n={\frac {1}{3}}=0{,}{\overline {3}}}$ ${\displaystyle n={\frac {1}{13}}=0{,}076923076923\ldots =0{,}{\overline {076923}}}$ ${\displaystyle 2\cdot {\frac {1}{13}}=0{,}153846153846\ldots =0{,}{\overline {153846}}}$ ${\displaystyle n={\frac {1}{7}}=0{,}14285714285\ldots =0{,}{\overline {142857}}}$ ${\displaystyle 10^{\text{Periodenlänge}}}$ ${\displaystyle 10^{6}}$ ${\displaystyle 142857{,}{\overline {142857}}}$ ${\displaystyle {\frac {1}{7}}=0{,}{\overline {142857}}}$ ${\displaystyle x=142857}$ ${\displaystyle k=2,3,4,\ldots ,n-1}$

${\displaystyle 2\cdot 142857=\;\;285714}$

${\displaystyle 3\cdot 142857=428571}$

${\displaystyle 4\cdot 142857=\;\;\;\;\;\,571428}$

${\displaystyle 5\cdot 142857=\;\;\;\;\;\;\;\,714285}$

${\displaystyle 6\cdot 142857=\;\;\;\,857142}$

Each time you get the same digits in the same order, just interchanged cyclically . Such numbers are called cyclic numbers . Above all, however, the period length of the fractional number was , therefore, maximally long, if you consider that division by at most different remainders not equal to zero can result (namely ). If the division resulted in a remainder, the decimal number and thus also the period would end (and would therefore not be a period because it just ends). In this respect, the term “long prime number” makes sense, because with a fraction the period length is and is therefore maximally long. In the case of composite numbers , the period length is never , so you can restrict yourself to prime numbers. ${\displaystyle n={\frac {1}{7}}={\frac {1}{p}}}$ ${\displaystyle p-1=6}$ ${\displaystyle p=7}$ ${\displaystyle p-1=6}$ ${\displaystyle 1,2,\ldots ,6}$ ${\displaystyle p}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle p-1}$ ${\displaystyle n}$ ${\displaystyle n-1}$

Abstract in the decimal system

In the decimal system you have the base . Take a prime number that is not a divisor of the base (i.e. and ) and form the fraction . This fraction should have the length of the period . Now you multiply this fraction by and subtract the original fraction so that the period after the decimal point disappears. The number corresponding to the number p is obtained . Is a cyclic number, it is a long prime number. ${\displaystyle b=10}$ ${\displaystyle p}$ ${\displaystyle b=10}$ ${\displaystyle p\not =2}$ ${\displaystyle p\not =5}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle p-1}$ ${\displaystyle b^{\text{Periodenlänge}}=10^{p-1}}$ ${\displaystyle x=10^{p-1}\cdot {\frac {1}{p}}-{\frac {1}{p}}={\frac {10^{p-1}-1}{p}}}$ ${\displaystyle x}$ ${\displaystyle p}$

Examples

• The following applies to the prime number in the decimal system (with base ): it is not a divisor of the base and it is a cyclic number, as was shown before. ${\displaystyle p=7}$ ${\displaystyle b=10}$ ${\displaystyle p=7}$ ${\displaystyle b=10}$ ${\displaystyle {\frac {b^{p-1}-1}{p}}={\frac {10^{7-1}-1}{7}}={\frac {999999}{7}}=142857}$
• The first long primes in the decimal system are the following:

7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, ... (sequence A001913 in OEIS )

• The periods of the reciprocal values ​​of the above long prime numbers (bis ) are as follows: ${\displaystyle p=59}$

142857, 5882352941176470, 526315789473684210, 4347826086956521739130, 3448275862068965517241379310, 2127659574468085106382978723404255319148936170, 1694915254237288135593220338983050847457627118644067796610 ... (sequence A004042 in OEIS )

• The number of long primes in the decimal system smaller than for are the following: ${\displaystyle 10^{n}}$ ${\displaystyle n=1,2,\ldots }$

1, 9, 60, 467, 3617, 29500, 248881, 2155288, 19016617, 170169241, 1539964486, 14063663530, 129413160100, ... (series A086018 in OEIS )

Example:

In the list above you can see the number at the 7th position . This means that there are long prime numbers below exactly . ${\displaystyle 248881}$ ${\displaystyle 10^{7}=10.000.000}$ ${\displaystyle 248881}$

properties

• The following four statements are synonymous:

• The number is a long prime number in the decimal system. ${\displaystyle p\in \mathbb {P} }$
• The cyclic number corresponding to the number has exact digits. ${\displaystyle p}$ ${\displaystyle x={\frac {10^{p-1}-1}{p}}}$ ${\displaystyle p-1}$
• For every remainder class there is one , so is. ${\displaystyle n=1,2,\ldots ,p-1}$ ${\displaystyle i}$ ${\displaystyle 10^{i}\equiv n{\pmod {p}}}$
• ${\displaystyle 10}$is a primitive root modulo ${\displaystyle p}$

• Let be a long prime number in the decimal system, which has a in the units position ( i.e. it has the form with ). Then: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle 1}$ ${\displaystyle p}$ ${\displaystyle p=10k+1}$ ${\displaystyle k\in \mathbb {N} }$

• Each digit appears the same number of times in the period from . ${\displaystyle 0,1,2,\ldots ,9}$ ${\displaystyle {\frac {1}{p}}}$
• The period length of is divisible by an integer. ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle 10}$

Example 1:

The following applies to the long prime number : The period length of the number is . In this period the number is included exactly times, as are the numbers . ${\displaystyle p=2741}$ ${\displaystyle {\frac {1}{p}}={\frac {1}{2741}}}$ ${\displaystyle p-1=2740}$ ${\displaystyle 0}$ ${\displaystyle 274}$ ${\displaystyle 1,2,3,\ldots 9}$

Example 2:

The smallest long prime numbers with in the ones place are the following: ${\displaystyle 1}$

61, 131, 181, 461, 491, 541, 571, 701, 811, 821, 941, 971, 1021, 1051, 1091, 1171, 1181, 1291, 1301, 1349, 1381, 1531, 1571, 1621, 1741, 1811, 1829, 1861, 2141, 2221, 2251, 2341, 2371, 2411, 2621, 2731, 2741, 2851, 2861, 2971, 3011, 3221, 3251, 3301, ... (series A073761 in OEIS )

• In the decimal system, the following prime numbers can never be long prime numbers:

• ${\displaystyle p=40k+n}$ With ${\displaystyle n\in \{1,3,9,13,27,31,37,39\}}$

• Studies have shown that in the decimal system about two thirds of the prime numbers with the following form are long prime numbers:

• ${\displaystyle p=40k+n}$ With ${\displaystyle n\in \{7,11,17,19,21,23,29,33\}}$

• ${\displaystyle 285}$of the primes of the form with are long primes. The first prime of this form that is not a long prime is . ${\displaystyle 295}$ ${\displaystyle p=120k+23}$ ${\displaystyle p<100000}$ ${\displaystyle p=20903}$
• A necessary, but not sufficient, condition that a long prime number is in the decimal system specifies the following property: ${\displaystyle p}$

The number is divisible by. ${\displaystyle 10^{p-1}-1=9\cdot R_{p-1}}$ ${\displaystyle p}$

(Here is a repunit , i.e. a number that consists exclusively of ones (and it has exactly places).) ${\displaystyle R_{p}}$ ${\displaystyle p}$

Example:

The following are dividers of : ${\displaystyle n}$ ${\displaystyle 10^{n-1}-1}$

1, 3, 7, 9, 11, 13, 17, 19, 23, 29, 31, 33, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 259, 263, ... ( continuation A104381 in OEIS )

(that is, all long primes must be in this list, but not all of these are long primes)

Unsolved problems

• It is believed that roughly all prime numbers in the decimal system are long prime numbers. (Follow A005596 in OEIS ) ${\displaystyle 37{,}39558136\ldots \,\%}$

(The number is called Artin's constant and actually refers to the number of prime numbers for which a primitive root is modulo . The conjecture is called Artin's conjecture and was first mentioned by Emil Artin .) ${\displaystyle 0{,}3739558136\ldots }$ ${\displaystyle 10}$ ${\displaystyle p}$

Example:

The following two lists together indicate what percentage of all prime numbers are under long prime numbers. First the numerator of this value: ${\displaystyle 10^{n}}$

1, 9, 5, 467, 3617, 14750, 248881, 2155288, 19016617, 170169241, ... (follow A103362 in OEIS )

The denominator of this value follows:

4, 25, 14, 1229, 9592, 39249, 664579, 5761455, 50847534, 455052511, ... (follow A103363 in OEIS )

In both of the above lists, the numbers and can be found in the 4th position . This means that all prime to long prime numbers are, which is very close to Artin's constant. Below are all prime numbers, which brings us even closer to Artin's constant. ${\displaystyle 467}$ ${\displaystyle 1229}$ ${\displaystyle {\frac {467}{1229}}\approx 37{,}998\,\%}$ ${\displaystyle 10^{4}}$ ${\displaystyle 10^{10}}$ ${\displaystyle {\frac {170169241}{455052511}}\approx 37{,}374\,\%}$

Long prime numbers in the dual system

Examples

• A long prime number in the dual system is the number . ${\displaystyle p=11=8+2+1={\underline {1}}\cdot 2^{3}+{\underline {0}}\cdot 2^{2}+{\underline {1}}\cdot 2^{1}+{\underline {1}}\cdot 2^{0}=1011_{2}}$

It is the number corresponding to the number the number . This number is cyclic in the dual system because: ${\displaystyle p=11}$ ${\displaystyle x={\frac {b^{p-1}-1}{p}}={\frac {2^{10}-1}{11}}={\frac {1023}{11}}=93=0001011101_{2}}$

${\displaystyle 2\cdot 93=0010_{2}\cdot 0001011101_{2}=0010111010_{2}}$.

${\displaystyle 3\cdot 93=0011_{2}\cdot 0001011101_{2}=0100010111_{2}}$.

${\displaystyle 4\cdot 93=0100_{2}\cdot 0001011101_{2}=0101110100_{2}}$.

${\displaystyle 5\cdot 93=0101_{2}\cdot 0001011101_{2}=0111010001_{2}}$.

${\displaystyle 6\cdot 93=0110_{2}\cdot 0001011101_{2}=1000101110_{2}}$.

${\displaystyle 7\cdot 93=0111_{2}\cdot 0001011101_{2}=1010001011_{2}}$.

${\displaystyle 8\cdot 93=1000_{2}\cdot 0001011101_{2}=1011101000_{2}}$.

${\displaystyle 9\cdot 93=1001_{2}\cdot 0001011101_{2}=1101000101_{2}}$.

${\displaystyle 10\cdot 93=1010_{2}\cdot 0001011101_{2}=1110100010_{2}}$.

The sequence of digits is cyclically exchanged each time when multiplying with (it is ), which is why there is a cyclic number in the dual system , therefore a long prime number in the dual system (by the way, not in the decimal system). ${\displaystyle x=93=0001011101_{2}}$ ${\displaystyle k=2,3,\ldots ,10}$ ${\displaystyle p-1=10}$ ${\displaystyle x=93=0001011101_{2}}$ ${\displaystyle p=11=1011_{2}}$

• The first long prime numbers in the dual system are the following:

3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797, ... (sequence A001122 in OEIS )

properties

• Let be a long prime number in the dual system (i.e. with a base ). Then: ${\displaystyle p}$ ${\displaystyle n=2}$

• For every remainder class there is one , so is. ${\displaystyle n=1,2,\ldots ,p-1}$ ${\displaystyle i}$ ${\displaystyle 2^{i}\equiv n{\pmod {p}}}$
• ${\displaystyle 2}$ is a primitive root modulo ${\displaystyle p}$

(This sentence is a special case of a sentence already mentioned above.)

• In the dual system, the following prime numbers can never be long prime numbers:

${\displaystyle p=8k+n}$ With ${\displaystyle n\in \{1,7\}}$

Proof:

For these , a quadratic remainder is modulo , so there must be a divisor of . The period length of in the dual system must divide and therefore cannot be. Thus there can not be a long prime number in the dual system. ${\displaystyle p}$ ${\displaystyle 2}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle 2^{\frac {p-1}{2}}-1}$ ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle {\frac {p-1}{2}}}$ ${\displaystyle p-1}$ ${\displaystyle p}$ ${\displaystyle \Box }$

• Let be a long prime number in the dual system. Then: ${\displaystyle p}$

${\displaystyle p}$has the form or with ${\displaystyle p=8k+3}$ ${\displaystyle p=8k+5}$ ${\displaystyle k\in \mathbb {N} }$

• All safe prime numbers with (inclusive ) are long prime numbers in the dual system. ${\displaystyle p}$ ${\displaystyle p\equiv 3{\pmod {8}}}$ ${\displaystyle p=3}$

The smallest safe prime numbers with this property are the following: ${\displaystyle p<2000}$

3, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, ...

• Studies have shown that in the dual system, around three quarters of the prime numbers with the following form are long prime numbers:

• ${\displaystyle p=8k+n}$ With ${\displaystyle n\in \{3,5\}}$

Example:

There are prime numbers that satisfy or congruence . of them are long prime numbers as the base . That's about . ${\displaystyle 87}$ ${\displaystyle p<1000}$ ${\displaystyle p\equiv 3{\pmod {8}}}$ ${\displaystyle p\equiv 5{\pmod {8}}}$ ${\displaystyle 67}$ ${\displaystyle b=2}$ ${\displaystyle 77\,\%}$

• ${\displaystyle 1078}$of the primes of the form with are long primes in the dual system. The first prime of this form that is not a long prime is . ${\displaystyle 1206}$ ${\displaystyle p=24k+5}$ ${\displaystyle p<100000}$ ${\displaystyle p=1013}$

Unsolved problems

• Artin suspects that there are infinitely many long prime numbers in the dual system.
• It is assumed that in the dual system there are an infinite number of prime twins that only consist of long prime numbers.

Long primes to the base b

Examples

• The smallest long prime numbers for the base for are the following (where 0 means that no such prime number exists): ${\displaystyle b}$ ${\displaystyle b=1,2,\ldots }$

2, 3, 2, 0 , 2, 11, 2, 3, 2, 7, 2, 5, 2, 3, 2, 0 , 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 11, 2, 3, 2, 19, 2, 0 , 2, 3, 2, 7, 2, 5, 2, 3, 2, 11, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 19, 2, 3, 2, 0 , 2, 7, 2, 3, 2, 19, 2, 5, 2, 3, 2, 13, 2, 5, 2, 3, 2, 5, 2, 11, 2, 3, 2, 5, 2, 11, 2, 3, 2, 7, 2, 7, 2, 3, 2, ... ( Follow A056619 in OEIS )

Example 1:

In the list above, the number is in the 6th position . That is, the smallest long prime number is the base . ${\displaystyle 11}$ ${\displaystyle p=11}$ ${\displaystyle b=6}$

Example 2:

In the list above, the number is in the 4th position . That is, there is no prime that is a long prime as the base . ${\displaystyle 0}$ ${\displaystyle p}$ ${\displaystyle b=4}$

• The following is a list of the smallest long prime numbers for various bases : ${\displaystyle b}$

properties

• The following four statements are synonymous:

• The number is a long prime base number ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle b\in \mathbb {N} }$
• The cyclic number corresponding to the number has exact digits ${\displaystyle p}$ ${\displaystyle x={\frac {b^{p-1}-1}{p}}}$ ${\displaystyle p-1}$
• For every remainder class there is one , so is. ${\displaystyle n=1,2,\ldots ,p-1}$ ${\displaystyle i}$ ${\displaystyle b^{i}\equiv n{\pmod {p}}}$
• ${\displaystyle b}$is a primitive root modulo ${\displaystyle p}$

(This sentence is a generalization of a sentence in the decimal system above .)

• Let be a long prime number for the base , which has a in the units place ( so it has the form with ). Then: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle b}$ ${\displaystyle 1}$ ${\displaystyle p}$ ${\displaystyle p=b\cdot k+1}$ ${\displaystyle k\in \mathbb {N} }$

• Each digit appears the same number of times in the period from . ${\displaystyle 0,1,2,\ldots ,b-1}$ ${\displaystyle {\frac {1}{p}}}$
• The period length of is divisible by an integer. ${\displaystyle {\frac {1}{p}}}$ ${\displaystyle b}$

(This sentence, too, is a generalization of a sentence in the decimal system above .)

• Every long base prime ends with or . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle b=12}$ ${\displaystyle 5}$ ${\displaystyle 7}$

(So ​​there are no long prime numbers for the base , which have a ones place.) ${\displaystyle b=12}$ ${\displaystyle 1}$

• Be a long prime base with or . Then: ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle b}$ ${\displaystyle b\equiv 0{\pmod {4}}}$ ${\displaystyle b\equiv 1{\pmod {4}}}$

There are no long prime numbers for the base which have a ones place. ${\displaystyle b}$ ${\displaystyle 1}$

(This sentence is a generalization of the sentence immediately above.)

Unsolved problems

• It is conjectured (also by Artin ) that there are infinitely many long prime numbers if the base is not a square number . ${\displaystyle b}$
• Let the basis not be a power of an integer (i.e. with ) and not be a basis whose square-free part is. Then it is assumed (also by Artin ): ${\displaystyle b}$ ${\displaystyle b\not =x^{y}}$ ${\displaystyle x,y\in \mathbb {N} }$ ${\displaystyle b}$ ${\displaystyle \equiv 1{\pmod {4}}}$

${\displaystyle 37{,}39558136\ldots \,\%}$of all prime numbers for the base are long prime numbers. ${\displaystyle b}$

Example:

The following numbers are not a power of an integer and do not have a square-free part, which is: ${\displaystyle \equiv 1{\pmod {4}}}$

2, 3, 6, 7, 10, 11, 12, 14, 15, 18, 19, 22, 23, 24, 26, 28, 30, 31, 34, 35, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 54, 55, 56, 58, 59, 60, 62, 63, 66, 67, 70, 71, 72, 74, 75, 76, 78, 79, 82, 83, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 99, 102, 103, 104, 106, 107, 108, ... (follow A085397 in OEIS )

generalization

A long prime number of degree n with base b is a prime number with the following property: ${\displaystyle p\in \mathbb {P} }$

Be with . Then: ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle 1\leq k\leq p-1}$

${\displaystyle {\frac {k}{p}}}$has different cycles in the corresponding decimal fraction expansion ${\displaystyle n}$

Examples in the decimal system

• Be and the base . Then: ${\displaystyle p=13}$ ${\displaystyle b=10}$

${\displaystyle {\begin{array}{cccccccc}1/13=0{,}{\overline {076923}}&2/13=0{,}{\overline {153846}}\\3/13=0{,}{\overline {230769}}&5/13=0{,}{\overline {384615}}\\4/13=0{,}{\overline {307692}}&6/13=0{,}{\overline {461538}}\\9/13=0{,}{\overline {692307}}&7/13=0{,}{\overline {538461}}\\10/13=0{,}{\overline {769230}}&8/13=0{,}{\overline {615384}}\\12/13=0{,}{\overline {923076}}&11/13=0{,}{\overline {846153}}\end{array}}}$

All 12 periods of (with ) are cyclical shifts of the first two periods. Thus the number has exactly different cycles and is therefore a long prime number of the 2nd degree as the base . ${\displaystyle {\frac {k}{13}}}$ ${\displaystyle k=1,2,\ldots ,12}$ ${\displaystyle p=13}$ ${\displaystyle 2}$ ${\displaystyle p=13}$ ${\displaystyle b=10}$

• Be and the base . Then: ${\displaystyle p=41}$ ${\displaystyle b=10}$

${\displaystyle {\begin{array}{cccccccc}1/41=0{,}{\overline {02439}}&2/41=0{,}{\overline {04878}}&3/41=0{,}{\overline {07317}}&4/41=0{,}{\overline {09756}}&5/41=0{,}{\overline {12195}}&6/41=0{,}{\overline {14634}}&11/41=0{,}{\overline {26829}}&15/41=0{,}{\overline {36585}}\\10/41=0{,}{\overline {24390}}&20/41=0{,}{\overline {48780}}&7/41=0{,}{\overline {17073}}&23/41=0{,}{\overline {56097}}&8/41=0{,}{\overline {19512}}&14/41=0{,}{\overline {34146}}&12/41=0{,}{\overline {29268}}&22/41=0{,}{\overline {53658}}\\16/41=0{,}{\overline {39024}}&32/41=0{,}{\overline {78048}}&13/41=0{,}{\overline {31707}}&25/41=0{,}{\overline {60975}}&9/41=0{,}{\overline {21951}}&17/41=0{,}{\overline {41463}}&28/41=0{,}{\overline {68292}}&24/41=0{,}{\overline {58536}}\\18/41=0{,}{\overline {43902}}&33/41=0{,}{\overline {80487}}&29/41=0{,}{\overline {70731}}&31/41=0{,}{\overline {75609}}&21/41=0{,}{\overline {51219}}&19/41=0{,}{\overline {46341}}&34/41=0{,}{\overline {82926}}&27/41=0{,}{\overline {65853}}\\37/41=0{,}{\overline {90243}}&36/41=0{,}{\overline {87804}}&30/41=0{,}{\overline {73170}}&40/41=0{,}{\overline {97560}}&39/41=0{,}{\overline {95121}}&26/41=0{,}{\overline {63414}}&38/41=0{,}{\overline {92682}}&35/41=0{,}{\overline {85365}}\end{array}}}$

All 40 periods of (with ) are cyclical shifts of eight different periods. Thus the number has exactly different cycles and is therefore a long prime number 8th degree as the base . ${\displaystyle {\frac {k}{41}}}$ ${\displaystyle k=1,2,\ldots ,40}$ ${\displaystyle p=41}$ ${\displaystyle 8}$ ${\displaystyle p=41}$ ${\displaystyle b=10}$

• In the decimal system, the first long prime numbers of the nth degree are the following (for ): ${\displaystyle n=1,2,\ldots }$

7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 859, 239, 27581, 9613, 18131, 13757, 33931, 9161, 118901, 6763, 18233, ... (sequence A054471 in OEIS )

Example:

The number is in the 8th position of the list above . This means that the smallest long prime number is 8th degree (in the decimal system). This prime number was used as an example directly above. ${\displaystyle 41}$ ${\displaystyle p=41}$

• In the dual system, the first long prime numbers of the nth degree are the following (for ): ${\displaystyle n=1,2,\ldots }$

3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257, 1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857, 72269, 3391, 683, 2593, 17029, 2687, 42701, 11161, 13099, 1103, 71293, 13121, 17467, 2143, 83077, 25609, 5581, ... (sequence A101208 in OEIS )

Example:

The number is in the 2nd position in the list above . That means that the smallest long prime number is 2nd degree (in the dual system). ${\displaystyle 7}$ ${\displaystyle p=7}$

• The following is a list of nth degree long prime numbers in the decimal system:

n	long prime numbers of degree n in the decimal system	OEIS episode
1	7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, ...	Follow A006883 in OEIS
2	3, 13, 31, 43, 67, 71, 83, 89, 107, 151, 157, 163, 191, 197, 199, 227, 283, 293, 307, 311, 347, 359, 373, 401, 409, 431, 439, 443, 467, 479, 523, 557, 563, 569, 587, 599, ...	Follow A275081 in OEIS
3	103, 127, 139, 331, 349, 421, 457, 463, 607, 661, 673, 691, 739, 829, 967, 1657, 1669, 1699, 1753, 1993, 2011, 2131, 2287, 2647, 2659, 2749, 2953, 3217, 3229, 3583, 3691, 3697, 3739, 3793, 3823, 3931, ...	Follow A055628 in OEIS
4th	53, 173, 277, 317, 397, 769, 773, 797, 809, 853, 1009, 1013, 1093, 1493, 1613, 1637, 1693, 1721, 2129, 2213, 2333, 2477, 2521, 2557, 2729, 2797, 2837, 3329, 3373, 3517, 3637, 3733, 3797, 3853, 3877, ...	Follow A056157 in OEIS
5	11, 251, 1061, 1451, 1901, 1931, 2381, 3181, 3491, 3851, 4621, 4861, 5261, 6101, 6491, 6581, 6781, 7331, 8101, 9941, 10331, 10771, 11251, 11261, 11411, 12301, 14051, 14221, 14411, ...	Follow A056210 in OEIS
6th	79, 547, 643, 751, 907, 997, 1201, 1213, 1237, 1249, 1483, 1489, 1627, 1723, 1747, 1831, 1879, 1987, 2053, 2551, 2683, 3049, 3253, 3319, 3613, 3919, 4159, 4507, 4519, 4801, 4813, 4831, 4969, ...	Follow A056211 in OEIS
7th	211, 617, 1499, 2087, 2857, 6007, 6469, 7127, 7211, 7589, 9661, 10193, 13259, 13553, 14771, 18047, 18257, 19937, 20903, 21379, 23549, 26153, 27259, 27539, 32299, 33181, 33461, 34847, 35491, 35897, ...	Follow A056212 in OEIS
8th	41, 241, 1601, 1609, 2441, 2969, 3041, 3449, 3929, 4001, 4409, 5009, 6089, 6521, 6841, 8161, 8329, 8609, 9001, 9041, 9929, 13001, 13241, 14081, 14929, 16001, 16481, 17489, 17881, 18121, 19001, ...	Follow A056213 in OEIS
9	73, 1423, 1459, 2377, 2503, 3457, 7741, 9433, 10891, 10909, 16057, 17299, 17623, 20269, 21313, 22699, 24103, 26263, 28621, 28927, 29629, 30817, 32257, 34273, 34327, ...	Follow A056214 in OEIS
10	281, 521, 1031, 1951, 2281, 2311, 2591, 3671, 5471, 5711, 6791, 7481, 8111, 8681, 8761, 9281, 9551, 10601, 11321, 12401, 13151, 13591, 14831, 14951, 15671, 16111, 16361, 18671, ...	Follow A056215 in OEIS

• The following is a list of long prime numbers of the nth degree in the dual system:

n	long prime numbers of the nth degree in the dual system	OEIS episode
1	3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, ...	Follow A001122 in OEIS
2	7, 17, 23, 41, 47, 71, 79, 97, 103, 137, 167, 191, 193, 199, 239, 263, 271, 311, 313, 359, 367, 383, 401, 409, 449, 463, 479, 487, 503, 521, 569, 599, 607, 647, 719, 743, 751, 761, 769, ...	Follow A115591 in OEIS
3	43, 109, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2749, 2917, ...	Follow A001133 in OEIS
4th	113, 281, 353, 577, 593, 617, 1033, 1049, 1097, 1153, 1193, 1201, 1481, 1601, 1889, 2129, 2273, 2393, 2473, 3049, 3089, 3137, 3217, 3313, 3529, 3673, 3833, 4001, 4217, 4289, 4457, 4801, 4817, 4937, ...	Follow A001134 in OEIS
5	251, 571, 971, 1181, 1811, 2011, 2381, 2411, 3221, 3251, 3301, 3821, 4211, 4861, 4931, 5021, 5381, 5861, 6221, 6571, 6581, 8461, 8501, 9091, 9461, 10061, 10211, 10781, 11251, 11701, 11941, 12541, ...	Follow A001135 in OEIS
6th	31, 223, 433, 439, 457, 727, 919, 1327, 1399, 1423, 1471, 1831, 1999, 2017, 2287, 2383, 2671, 2767, 2791, 2953, 3271, 3343, 3457, 3463, 3607, 3631, 3823, 3889, 4129, 4423, 4519, 4567, 4663, 4729, 4759, ...	Follow A001136 in OEIS
7th	1163, 1709, 2003, 3109, 3389, 3739, 5237, 5531, 5867, 7309, 9157, 9829, 10627, 10739, 11117, 11243, 11299, 11411, 11467, 13259, 18803, 20147, 20483, 21323, 21757, 27749, 27763, 29947, ...	Follow A152307 in OEIS
8th	73, 89, 233, 937, 1217, 1249, 1289, 1433, 1553, 1609, 1721, 1913, 2441, 2969, 3257, 3449, 4049, 4201, 4273, 4297, 4409, 4481, 4993, 5081, 5297, 5689, 6089, 6449, 6481, 6689, 6857, 7121, 7529, 7993, ...	Follow A152308 in OEIS
9	397, 7867, 10243, 10333, 12853, 13789, 14149, 14293, 14563, 15643, 17659, 18379, 18541, 21277, 21997, 23059, 23203, 26731, 27739, 29179, 29683, 31771, 34147, 35461, 35803, 36541, 37747, 39979, ...	Follow A152309 in OEIS
10	151, 241, 431, 641, 911, 3881, 4751, 4871, 5441, 5471, 5641, 5711, 6791, 6871, 8831, 9041, 9431, 10711, 12721, 13751, 14071, 14431, 14591, 15551, 16631, 16871, 17231, 17681, 17791, 18401, 19031, 19471, ...	Follow A152310 in OEIS

Individual evidence

1. ^ ^{A b} Leonard Eugene Dickson : History of the Theory of Numbers, Volume 1 - Divisibility and primality. Carnegie Institution of Washington , 1919, p. 166 , accessed July 11, 2018 . 
2. ^ John H. Conway , Richard K. Guy : The Book of Numbers. Springer-Verlag, 1996, pp. 157–163, 166–171 , accessed on July 11, 2018 (English). 
3. ↑ ^{a b} Eric W. Weisstein : Full Reptend Prime . In: MathWorld (English).
4. ↑ ^{a b} Neil Sloane : Primes with primitive root 2 - Comments. OEIS , accessed July 12, 2018 . 

Web links

• Eric W. Weisstein : Full Reptend Prime . In: MathWorld (English).
• Eric W. Weisstein : Artins Constant . In: MathWorld (English).
• Full Reptend Prime . In: PlanetMath . (English)