Reciprocals of primes
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737.
Like all rational numbers, the reciprocals of primes have repeating decimal representations. In his later years, George Salmon (1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes.[1]
Contemporaneously, William Shanks (1812–1882) calculated numerous reciprocals of primes and their repeating periods, and published two papers "On Periods in the Reciprocals of Primes" in 1873[2] and 1874.[3] In 1874 he also published a table of primes, and the periods of their reciprocals, up to 20,000 (with help from and "communicated by the Rev. George Salmon"), and pointed out the errors in previous tables by three other authors.[4]
Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878.[5] For a prime p, the period of its reciprocal divides p − 1.[6]
The sequence of recurrence periods of the reciprocal primes (sequence A002371 in the OEIS) appears in the 1973 Handbook of Integer Sequences.
List of reciprocals of primes
| Prime (p) |
Period length |
Reciprocal (1/p) |
|---|---|---|
| 2 | 0 | 0.5 |
| 3 | † 1 | 0.3 |
| 5 | 0 | 0.2 |
| 7 | * 6 | 0.142857 |
| 11 | † 2 | 0.09 |
| 13 | 6 | 0.076923 |
| 17 | * 16 | 0.0588235294117647 |
| 19 | * 18 | 0.052631578947368421 |
| 23 | * 22 | 0.0434782608695652173913 |
| 29 | * 28 | 0.0344827586206896551724137931 |
| 31 | 15 | 0.032258064516129 |
| 37 | † 3 | 0.027 |
| 41 | 5 | 0.02439 |
| 43 | 21 | 0.023255813953488372093 |
| 47 | * 46 | 0.0212765957446808510638297872340425531914893617 |
| 53 | 13 | 0.0188679245283 |
| 59 | * 58 | 0.0169491525423728813559322033898305084745762711864406779661 |
| 61 | * 60 | 0.016393442622950819672131147540983606557377049180327868852459 |
| 67 | 33 | 0.014925373134328358208955223880597 |
| 71 | 35 | 0.01408450704225352112676056338028169 |
| 73 | 8 | 0.01369863 |
| 79 | 13 | 0.0126582278481 |
| 83 | 41 | 0.01204819277108433734939759036144578313253 |
| 89 | 44 | 0.01123595505617977528089887640449438202247191 |
| 97 | * 96 | 0.010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567 |
| 101 | † 4 | 0.0099 |
| 103 | 34 | 0.0097087378640776699029126213592233 |
| 107 | 53 | 0.00934579439252336448598130841121495327102803738317757 |
| 109 | * 108 | 0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211 |
| 113 | * 112 | 0.0088495575221238938053097345132743362831858407079646017699115044247787610619469026548672566371681415929203539823 |
* Full reptend primes are italicised.
† Unique primes are highlighted.
Full reptend primes
A full reptend prime, full repetend prime, proper prime[7]:166 or long prime in base b is an odd prime number p such that the Fermat quotient
- [math]\displaystyle{ q_p(b) = \frac{b^{p - 1} - 1}{p} }[/math]
(where p does not divide b) gives a cyclic number with p − 1 digits. Therefore, the base b expansion of [math]\displaystyle{ 1/p }[/math] repeats the digits of the corresponding cyclic number infinitely.
Unique primes
A prime p (where p ≠ 2, 5 when working in base 10) 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.[8] 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. The next larger unique prime is 9091 with period 10, though the next larger period is 9 (its prime being 333667). Unique primes were described by Samuel Yates in 1980.[9] A prime number p is unique if and only if there exists an n such that
- [math]\displaystyle{ \frac{\Phi_n(10)}{\gcd(\Phi_n(10), n)} }[/math]
is a power of p, where [math]\displaystyle{ \Phi_n(b) }[/math] denotes the [math]\displaystyle{ n }[/math]th cyclotomic polynomial evaluated at [math]\displaystyle{ b }[/math]. The value of n is then the period of the decimal expansion of 1/p.[10]
At present, more than fifty decimal unique primes or probable primes are known. However, there are only twenty-three unique primes below 10100.
List of decimal unique primes
The following table lists the first 23 unique primes in decimal (sequence A040017 in the OEIS).
| Script error: No such module "Vertical header". | 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 |
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.
Further down, repunit prime [math]\displaystyle{ R_{{317}} }[/math] is the 29th unique prime, and [math]\displaystyle{ R_{{1031}} }[/math] the 45th.
Where A040017 contains a list of unique primes, A007615 are those primes ordered by period length; A051627 contains periods (ordered by corresponding primes) and A007498 contains periods, sorted, corresponding with A007615.
Largest unique primes
In 1996 the largest proven unique prime was (101132 + 1)/10001 or, using the notation above, (99990000)141 + 1. It has 1128 digits;[11] this record has been improved many times since then.
(As of 2023), the largest proven unique prime is [math]\displaystyle{ R_{{86453}} }[/math], which repeats 86453 digits.[12] On the other hand, repunit (108177207 – 1) / 9 is the largest known probable unique prime.[13]
Generalized unique primes
A unique prime p fulfilling the standard definition in another base b > 1 is called a generalized unique prime.[10]
The largest known generalized unique prime (discovered October 2023) is [math]\displaystyle{ \Phi_3(-516693^{1048576}) = 516693^{2097152}-516693^{1048576}+1 }[/math],[10][14][15] which is the seventh largest known prime of any type, and the largest known non-Mersenne prime (as of January 2024).[16]
References
- ↑ "Obituary Notices – George Salmon". Proceedings of the London Mathematical Society. Second Series 1: xxii–xxviii. 1904. https://books.google.com/books?id=IPo7AQAAMAAJ&pg=PR28. Retrieved 27 March 2022. "...there was one branch of calculation which had a great fascination for him. It was the determination of the number of figures in the recurring periods in the reciprocals of prime numbers.".
- ↑ Shanks, William (1873). "On Periods in the Reciprocals of Primes". The Messenger of Mathematics II: 41–43. https://books.google.com/books?id=EbJYAAAAcAAJ&pg=PA41&. Retrieved 27 March 2022.
- ↑ Shanks, William (1874). "On Periods in the Reciprocals of Primes". The Messenger of Mathematics III: 52–55. https://books.google.com/books?id=EfPxAAAAMAAJ&pg=PA52. Retrieved 27 March 2022.
- ↑ Shanks, William (1874). "On the Number of Figures in the Period of the Reciprocal of Every Prime Number Below 20,000". Proceedings of the Royal Society of London 22: 200–210. https://www.jstor.org/stable/112821. Retrieved 27 March 2022.
- ↑ Glaisher, J. W. L. (1878). "On circulating decimals with special reference to Henry Goodwin's 'Table of circles' and 'Tabular series of decimal quotients'". Proceedings of the Cambridge Philosophical Society: Mathematical and physical sciences 3 (V): 185–206. https://books.google.com/books?id=juJUAAAAYAAJ&pg=PA185. Retrieved 27 March 2022.
- ↑ Cook, John D.. "Reciprocals of primes". https://www.johndcook.com/blog/2018/05/10/reciprocals-of-primes/#:~:text=Reciprocals%20of%20primes&text=For%20any%20prime%20p%20except,a%20divisor%20of%20p%2D1..
- ↑ Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
- ↑ Caldwell, Chris. "Unique prime". The Prime Pages. http://primes.utm.edu/glossary/xpage/UniquePrime.html.
- ↑ Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314.
- ↑ 10.0 10.1 10.2 "Generalized Unique". https://t5k.org/top20/page.php?id=44.
- ↑ "Wolfram Alpha". https://www.wolframalpha.com/input?i=%2810%5E1132%2B1%29%2F10001.
- ↑ The Top Twenty Unique; Chris Caldwell
- ↑ PRP Records: Probable Primes Top 10000
- ↑ "Phi(3, −5166931048576)". https://t5k.org/primes/page.php?id=136490.
- ↑ [math]\displaystyle{ \Phi_3(x)=x^2+x+1 }[/math], the 3rd cyclotomic polynomial
- ↑ https://t5k.org/largest.html#biggest
External links
- Parker, Matt (March 14, 2022). "The Reciprocals of Primes - Numberphile". https://www.youtube.com/watch?v=DmfxIhmGPP4.
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