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]

The last part of Shanks's 1874 table of primes and their repeating periods. In the top row, 6952 should be 6592 (the error is easy to find, since the period for a prime

p

must divide

p - 1

). In his report extending the table to 30,000 in the same year, Shanks did not report this error, but reported that in the same column, opposite 19841, the 1984 should be 64. *Another error which may have been corrected since his work was published is opposite 19423, the reciprocal repeats every 6474 digits, not every 3237.

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[edit]

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[edit]

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

q_{p}(b)={\frac {b^{p-1}-1}{p}}

(where p does not divide b) gives a cyclic number with p − 1 digits. Therefore, the base b expansion of $1/p$ repeats the digits of the corresponding cyclic number infinitely.

Unique primes[edit]

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

{\frac {\Phi _{n}(10)}{\gcd(\Phi _{n}(10),n)}}

is a power of p, where $\Phi _{n}(b)$ denotes the $n$ th cyclotomic polynomial evaluated at $b$ . 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[edit]

The following table lists the first 23 unique primes in decimal (sequence A040017 in the OEIS).

Period length	Prime
1	3
2	11
3	37
4	101
10	9,091
12	9,901
9	333,667
14	909,091
24	99,990,001
36	999,999,000,001
48	9,999,999,900,000,001
38	909,090,909,090,909,091
19	1,111,111,111,111,111,111
23	11,111,111,111,111,111,111,111
39	900,900,900,900,990,990,990,991
62	909,090,909,090,909,090,909,090,909,091
120	100,009,999,999,899,989,999,000,000,010,001
150	10,000,099,999,999,989,999,899,999,000,000,000,100,001
106	9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
93	900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991
134	909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
294	142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143
196	999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001

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 $R_{317}$ is the 29th unique prime, and $R_{1031}$ 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[edit]

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^[update], the largest proven unique prime is $R_{86453}$ , which repeats 86453 digits.^[12] On the other hand, repunit (10^8177207 – 1) / 9 is the largest known probable unique prime.^[13]

Generalized unique primes[edit]

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 $\Phi _{3}(-516693^{1048576})=516693^{2097152}-516693^{1048576}+1$ ,^[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[edit]

^ "Obituary Notices – George Salmon". Proceedings of the London Mathematical Society. Second Series. 1: xxii–xxviii. 1904. 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. Retrieved 27 March 2022.
^ Shanks, William (1874). "On Periods in the Reciprocals of Primes". The Messenger of Mathematics. III: 52–55. 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. 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. Retrieved 27 March 2022.
^ Cook, John D. "Reciprocals of primes". johndcook.com. Retrieved 6 April 2022.
^ Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.
^ Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014.
^ Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314. Zbl 0445.10009.
^ ^a ^b ^c "Generalized Unique". Prime Pages. Retrieved 9 December 2023.
^ "Wolfram Alpha". Wolfram Alpha. Retrieved 8 June 2023.
^ The Top Twenty Unique; Chris Caldwell
^ PRP Records: Probable Primes Top 10000
^ "Phi(3, −516693^1048576)". Prime Pages. Retrieved 22 December 2023.
^ $\Phi _{3}(x)=x^{2}+x+1$ , the 3rd cyclotomic polynomial
^ https://t5k.org/largest.html#biggest

External links[edit]

Parker, Matt (March 14, 2022). "The Reciprocals of Primes - Numberphile". YouTube.

[proc-1] "Obituary Notices – George Salmon". Proceedings of the London Mathematical Society. Second Series. 1: xxii–xxviii. 1904. 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.

[2] Shanks, William (1873). "On Periods in the Reciprocals of Primes". The Messenger of Mathematics. II: 41–43. Retrieved 27 March 2022.

[3] Shanks, William (1874). "On Periods in the Reciprocals of Primes". The Messenger of Mathematics. III: 52–55. Retrieved 27 March 2022.

[shanks20000-4] 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. Retrieved 27 March 2022.

[glaisher-5] 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. Retrieved 27 March 2022.

[cook-6] Cook, John D. "Reciprocals of primes". johndcook.com. Retrieved 6 April 2022.

[Dickson-7] Dickson, Leonard E., 1952, History of the Theory of Numbers, Volume 1, Chelsea Public. Co.

[8] Caldwell, Chris. "Unique prime". The Prime Pages. Retrieved 11 April 2014.

[9] Yates, Samuel (1980). "Periods of unique primes". Math. Mag. 53: 314. Zbl 0445.10009.

[top20gen-10] "Generalized Unique". Prime Pages. Retrieved 9 December 2023.

[11] "Wolfram Alpha". Wolfram Alpha. Retrieved 8 June 2023.

[12] The Top Twenty Unique; Chris Caldwell

[13] PRP Records: Probable Primes Top 10000

[14] "Phi(3, −516693^1048576)". Prime Pages. Retrieved 22 December 2023.

[15] $\Phi _{3}(x)=x^{2}+x+1$ , the 3rd cyclotomic polynomial

[16] ttps://t5k.org/largest.html#biggest

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v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers