Reciprocals of primes

Last updated May 12, 2024

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.

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

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

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

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

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

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

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]

Related Research Articles

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form $M n = 2 n - 1$ for some integer $n$ . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If $n$ is a composite number then so is $2 n - 1$ . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form $M p = 2 p - 1$ for some prime $p$ .

Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11" – this is called a standard form. A base-φ numeral that includes the digit sequence "11" can always be rewritten in standard form, using the algebraic properties of the base φ — most notably that φ¹ + φ⁰ = φ². For instance, 11_φ = 100_φ.

In mathematics, thenth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of $and is not a divisor of for any k < n . Its roots are all n th primitive roots of unity, where k runs over the positive integers not greater than n and coprime to n . In other words, the n th cyclotomic polynomial is equal to$

11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.

21 (twenty-one) is the natural number following 20 and preceding 22.

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.

In mathematics, a quadratic irrational number is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as

23 (twenty-three) is the natural number following 22 and preceding 24.

A power of two is a number of the form $2 n$ where $n$ is an integer, that is, the result of exponentiation with number two as the base and integer $n$ as the exponent.

84 (eighty-four) is the natural number following 83 and preceding 85.

In number theory, a Wagstaff prime is a prime number of the form

181 is the natural number following 180 and preceding 182.

In number theory, a full reptend prime, full repetend prime, proper prime or long prime in base b is an odd prime number p such that the Fermat quotient

271 is the natural number after 270 and before 272.

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic ; if this sequence consists only of zeros, the decimal is said to be terminating, and is not considered as repeating. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227/555, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is 593/53, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator $p$ and a non-zero denominator $q$ . For example, is a rational number, as is every integer. The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface $Q$ , or blackboard bold

In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to $Q$ , the field of rational numbers.

In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function.

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as $e$ and $π$ occurring in such diverse contexts as geometry, number theory, statistics, and calculus.

References

↑ "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.
1 2 3 "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

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

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[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] 1 2 3 "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