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Numbers that contain only the digit 1
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
Repunit
This is a list of repunit primes in various bases. Base-2 repunit primes are called Mersenne primes. The first few base-3 repunit primes are 13, 1093
List_of_repunit_primes
Natural number with a decimal representation made of repeated instances of the same digit
repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which
Repdigit
Prime number of the form 2^n – 1
11111111111111111111111, ... (sequence A004022 in the OEIS). These primes are called repunit primes. Another example is when we take b = −12, we get n values of: 2
Mersenne_prime
Symbols used to write numbers
faulty. Repunits are integers that are represented with only the digit 1. For example, 1111 (one thousand, one hundred and eleven) is a repunit. Repdigits
Numerical_digit
Natural number
squaring the square). 111 is R 3 {\displaystyle R_{3}} or the second repunit in decimal, a number like 11, 111, or 1111 that consists of repeated units
111_(number)
Natural number
051 = fifth Keith prime 1,089,270 = harmonic divisor number 1,111,111 = repunit 1,112,083 = logarithmic number 1,129,30832 + 1 is prime 1,136,689 = Pell
1,000,000
Natural number
factorisation of generalized repunits for which no generalized repunit primes are known. It is known that the generalized repunit number 185 p − 1 184 {\displaystyle
185_(number)
For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below: a = 3: 13, 1093, 797161
List_of_prime_numbers
Natural number
in decimal R 19 {\displaystyle R_{19}} is also the second to be a prime repunit (after R 2 {\displaystyle R_{2}} ), followed by R 23 {\displaystyle R_{23}}
23_(number)
Type of prime number
primes, but later they were also called absolute primes. In base 2, only repunits can be permutable primes, because any 0 permuted to the ones place results
Permutable_prime
Natural number
Eisenstein integers. 271 is the largest prime factor of the five-digit repunit 11111, and the largest prime number for which the decimal period of its
271_(number)
Prime number of the form (2ᵖ+1)/3
641, 2137, 3011, 268207, ... (sequence A001562 in the OEIS). See Repunit#Repunit primes for the list of the generalized Wagstaff primes base b {\displaystyle
Wagstaff_prime
Natural number
palindromic prime in 2 consecutive bases: 23 (KLK23) and 24 (J5J24) 11111 = Repunit 11297 = Number of planar partitions of 16 11298 = Riordan number 11311
10,000
Type of composite integer
(sequence A059754 in the OEIS). Smith numbers can be constructed from factored repunits.[verification needed] As of 2010[update], the largest known Smith number
Smith_number
Natural number
with index 2. a palindromic number in bases 7 (3137) and 12 (11112). a repunit in base 12, so it is a unique prime in the same base a prime whose digits
157_(number)
Natural number
divisible by the sum of its digits, making it a Harshad number. The smallest repunit probable prime in base 152 was found in June 2015, it has 589570 digits
152_(number)
Natural number
Sierpinski problem 55555 = repdigit 55860 = harmonic divisor number 55987 = repunit prime in base 6 56011 = Wedderburn-Etherington number 56092 = the number
50,000
Natural number
is a square (11 times 11) the sum of the powers of 3 from 0 to 4, so a repunit in ternary. Furthermore, 121 is the only square of the form 1 + p + p 2
121_(number)
Numeral ambigram
concept, professional mathematicians generally are not. Like the concept of repunits and palindromic numbers, the concept of strobogrammatic numbers is base-dependent
Strobogrammatic_number
may also be expressed as saying that there are only two numbers that are repunits with at least three digits in two different bases. The number 31 may be
Goormaghtigh_conjecture
Type of prime number
a repunit, a number consisting only of n ones (in base 10). There are no other circular primes up to 1025. The only other known examples are repunit primes
Circular_prime
Natural number
413,504 = 147 107,890,609 = Wedderburn-Etherington number 111,111,111 = repunit, square root of 12345678987654321 111,111,113 = Chen prime, Sophie Germain
100,000,000
Natural number
unlabeled nodes 110,075,314,176 = 3317762 = 5764 = 248 111,111,111,111 = repunit 118,587,876,497 = 49133 = 179 127,004,500,762 = number of parallelogram
100,000,000,000
Recursive integer sequence
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Catalan_number
Number with an integer power equal to 1
integer ≥ 2 for z, this sum becomes a base z repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime
Root_of_unity
Number divisible only by 1 and itself
Washington 2014, p. 41. For instance see Guy 2013, A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape k ⋅ 2 n + 1 {\displaystyle k\cdot 2^{n}+1}
Prime_number
heptagonal number 1031 = exponent and number of ones for the fifth base-10 repunit prime, Sophie Germain prime, super-prime, Chen prime 1032 = sum of two
1000_(number)
Natural number
prime and a star prime. It is also the smallest Wieferich prime. 1093 is a repunit prime in base 3 because: 1093 = 1111111 3 = 3 6 + 3 5 + 3 4 + 3 3 + 3 2
1093_(number)
Mathematics: (10109297 − 1)/9, with 109,297 digits, is the largest proven repunit prime in base 10 as of May 2025[update]. Mathematics: approximately 7.76
Orders_of_magnitude_(numbers)
Natural number
1038232 = 22093 = 476 11,019,960,576 = 1049762 = 3244 = 188 11,111,111,111 = repunit 11,123,060,678 = number of free 21-ominoes 11,874,568,703 = number of partitions
10,000,000,000
Indian recreational mathematician (1905–1986)
Demlo numbers 1, 121, 12321, 1234321, ..., which are the squares of the repunits 1, 11, 111,1111, .... Prahalad Chunnilal Vaidya "क्या आप जानते हैं जादुई
D._R._Kaprekar
Iterative algorithm on numbers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Kaprekar's_routine
Type of number in mathematics
found on 18 October 2021 by Ryan Propper and Serge Batalov. All decimal repunits are palindromic numbers, some of which are prime. Among them, the largest
Palindromic_prime
Numbers obtained by adding the two previous ones
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Fibonacci_sequence
Integer having a non-trivial divisor
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Composite_number
Natural number
= automorphic number 110,880 = 30th highly composite number 111,111 = repunit 111,777 = smallest natural number requiring 17 syllables in American English
100,000
Natural number
number. 317 is the exponent (and number of ones) in the fourth base-10 repunit prime. 319 = 11 × 29. 319 is the sum of three consecutive primes (103 +
300_(number)
Natural number
number of partially ordered set with 12 unlabeled elements 1,111,111,111 : repunit. 1,129,760,415 : 23rd Motzkin number. 1,134,903,170 : 45th Fibonacci number
1,000,000,000
Natural number
30537 = Riordan number 30694 = open meandric number 30941 = first base 13 repunit prime 31116 = octahedral number 31185 = number of partitions of 39 31337
30,000
power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b? For any given integers k ≥ 1 , b ≥ 2 , c ≠ 0 {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Natural number
over is allowed) where complements are equivalent 11,111,111,111,111 : repunit 11,258,999,739,560 : number of 50-bead binary necklaces with beads of 2
10,000,000,000,000
Arithmetic operation
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Exponentiation
Natural number
1/281 is 28. However, in binary, it has period length 70. The generalized repunit number 281 p − 1 280 {\displaystyle {\frac {281^{p}-1}{280}}} is composite
281_(number)
Natural number
22222 22447 = cuban prime 22527 = Woodall number: 11 × 211 − 1 22621 = repunit prime in base 12 22699 = one of five remaining Seventeen or Bust numbers
20,000
Natural number
787,776 : Leyland number using 4 & 20 (420 + 204) 1,111,111,111,111 : repunit 1,117,594,214,815 : 62nd perfect totient number 1,124,388,064,800 : 67th
1,000,000,000,000
Number equal to the sum of its proper divisors
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Perfect_number
Numbers with a certain property involving recursive summation
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Happy_number
Natural number
pair with 2783 (first definition) 2791 – cuban prime 2801 – first base 7 repunit prime 2803 – super-prime 2806 – centered pentagonal number, sum of the
2000_(number)
Ten raised to an integer power
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Power_of_10
Product of an integer with itself
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Square_number
Online database of integer sequences
"base" if defined as "primes of the form 2^n − 1". However, defined as "repunit primes in binary," the sequence would rate the keyword "base". bref – "sequence
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
2002 TK102 Repeated 1 1 Ceres 11 Parthenope 111 Ate 1111 Reinmuthia 11111 Repunit (111111) 2001 VO84 Repeated 2 2 Pallas 22 Kalliope 222 Lucia 2222 Lermontov
List_of_exceptional_asteroids
Integer filtered out using a sieve similar to that of Eratosthenes
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Lucky_number
Natural number
number using 6 & 9 (69 + 96) 10,976,184 = Logarithmic number 11,111,111 = Repunit 11,316,496 = 33642 = 584 11,390,625 = 33752 = 2253 = 156 11,405,773 = Leonardo
10,000,000
Figurate number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Triangular_number
Two raised to an integer power
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Power_of_two
Base-1 numeral system
to represent 5 as a tally. Unary numbers should be distinguished from repunits, which are also written as sequences of ones but have their usual decimal
Unary_numeral_system
Number that represents a hexagon with a dot in the center
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Centered_hexagonal_number
Type of Poulet number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Super-Poulet_number
Number, product of consecutive integers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Pronic_number
Number used for counting
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Natural_number
Type of integer sequence
a_{n}=2^{n}-1} form a strong divisibility sequence, which is Un(3, 2). The repunit numbers R(b) n for n = 1, 2, ... in any base b form a strong divisibility
Divisibility_sequence
Class of prime numbers
Springer-Verlag, 1996. Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers"; in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988)
Full_reptend_prime
Mathematical concept in prime numbers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Idoneal_number
Result of multiplying eight instances of a number together
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Eighth_power
Result on density of prime numbers
Good Super Higgs Highly cototient Unique Base-dependent Palindromic Emirp Repunit (10n − 1)/9 Permutable Circular Truncatable Minimal Delicate Primeval Full
Bertrand's_postulate
Natural number whose divisor sum is greater than that of any smaller number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Highly_abundant_number
Type of composite number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Giuga_number
not a perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known. 210 Smallest base such
List_of_numeral_systems
Variant of fast Fourier transform
Granger and Scott demonstrated using IBDWT-inspired "GRP (generalized repunit prime) multiplication" to accelerate eliptic curve cryptography over F(2521-1)
Irrational base discrete weighted transform
Irrational_base_discrete_weighted_transform
Number puzzle
and other "non-mathematical" approaches (e.g. palindromic numbers and repunits) where same result can be achieved through algebraic means. "Cross-figure
Cross-figure
American mathematician (1928–2019)
personal computers. He found many large prime numbers of special forms: repunits, Fibonacci primes, prime Lucas numbers, twin primes, Sophie Germain primes
Harvey_Dubner
Class of natural numbers with many divisors
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Superior highly composite number
Superior_highly_composite_number
Number that is the result of operation on its own digits
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Friedman_number
Abundant number whose proper divisors are all deficient numbers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Primitive_abundant_number
Number of points in an octagonal arrangement
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Octagonal_number
Type of number introduced by Mike Keith
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Keith_number
Numbers parameterizing ways to partition a set
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Type of composite number with an even number of digits
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Vampire_number
Count of the possible partitions of a set
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Bell_number
Type of figurate number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Nonagonal_number
Sequence of integers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Padovan_sequence
Infinite integer series where the next number is the sum of the two preceding it
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Lucas_number
— November 2, 1995 Kiyosato S. Otomo · 12 km MPC · JPL 11111 Repunit 1995 WL Repunit November 16, 1995 Ōizumi T. Kobayashi KOR 5.7 km MPC · JPL 11112
List of minor planets: 11001–12000
List_of_minor_planets:_11001–12000
Combinatorial sequence of numbers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Dedekind_number
Integer of the form 3 × 2^n – 1 for non-negative n
Good Super Higgs Highly cototient Unique Base-dependent Palindromic Emirp Repunit (10n − 1)/9 Permutable Circular Truncatable Minimal Delicate Primeval Full
Thabit_number
Numbers k where x - phi(x) = k has many solutions
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Highly_cototient_number
number Stoneham number Champernowne constant Absolutely normal number Repunit Repdigit Semiprime Almost prime Unique prime Factorial prime Permutable
List of recreational number theory topics
List_of_recreational_number_theory_topics
Sum of a number's digits
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Digit_sum
Integer having only small prime factors
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Smooth_number
Number, non-palindrome after repeated sum with reverse
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Lychrel_number
Type of figurate number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Hexagonal_number
Base-dependent property of integers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Kaprekar_number
Numbers with many divisors
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Highly_composite_number
Numbers that evenly divide powers of 60
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Regular_number
Three raised to an integer power
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Power_of_three
Square of a triangular number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Squared_triangular_number
Positive integer of the form (2^(2^n))+1
Good Super Higgs Highly cototient Unique Base-dependent Palindromic Emirp Repunit (10n − 1)/9 Permutable Circular Truncatable Minimal Delicate Primeval Full
Fermat_number
Composite number in number theory
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Carmichael_number
REPUNIT
REPUNIT
REPUNIT
REPUNIT
Boy/Male
Hindu, Indian
Writter
Boy/Male
Tamil
Joy
Girl/Female
Arabic, Muslim
Iron
Boy/Male
Indian, Sanskrit
Unalterable
Girl/Female
French, German, Hebrew
Dear; The Plain
Biblical
stranger; gathered together
Boy/Male
Hindu
God murugans name (son of Lord Shiva)
Girl/Female
Latin American
Victory; triumphant. Famous Bearer: Queen Victoria.
Girl/Female
German
meaning she whose singing lures men to destruction.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Telugu
Lord Shiva
REPUNIT
REPUNIT
REPUNIT
REPUNIT
REPUNIT