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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
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
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
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
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
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
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)
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
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)
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)
Type of prime number
A003459 in the OEIS) Where Rn := 10n−19 is a repunit, a number consisting only of n ones (in base 10). Any repunit prime is a permutable prime with the above
Permutable_prime
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)
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
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)
Natural number
121 is a square (11 × 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)
Natural number
numbers in the Sierpinski problem 55860 = harmonic divisor number 55987 = repunit prime in base 6 56011 = Wedderburn-Etherington number 56092 = the number
50,000
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
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
Mathematics: (10109297 − 1)/9, with 109,297 digits, is the largest proven repunit prime in base 10 as of May 2025[update]. Mathematics: 5262,144 ≈ 6.206069879×10183230
Orders_of_magnitude_(numbers)
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
Iterative algorithm on numbers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Kaprekar's_routine
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)
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
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)
Arithmetic operation
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Exponentiation
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⋅2n+1. pp. 13–21. "Record 12-Million-Digit
Prime_number
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
Recursive integer sequence
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Catalan_number
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
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
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
Natural number
unlabeled nodes 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
Chen prime. It is the exponent and number of ones for the fifth base-10 repunit prime. 1033 is a prime number and an emirp. It forms a twin prime pair
1000_(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
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
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
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
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
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
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
Sequence of integers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Padovan_sequence
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
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
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
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
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
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
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)
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
Figurate number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Triangular_number
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
Numbers with many divisors
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Highly_composite_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
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
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
Integer with a specific property
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Sierpiński_number
Combinatorial sequence of numbers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Dedekind_number
Mathematical formula
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Multiplicative_digital_root
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
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
Product of the first "n" prime numbers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Primorial
Result of multiplying four instances of a number together
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Fourth_power
Unsolved problem in mathematics
upper bound for numbers with the Lehmer property and its application to repunit numbers". International Journal of Number Theory. 15 (7): 1463–1468. ISSN 1793-0421
Lehmer's_totient_problem
Natural number
a sum of fewer than 548 ninth powers 19513 = tribonacci number 19531 = repunit prime in base 5 19600 = 1402, tetrahedral number 19601/13860 ≈ √2 19609
10,000
Numbers in a type of Lucas sequence
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Jacobsthal_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
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
Prime such that p^2 divides 2^(p-1)-1
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Wieferich_prime
Centered figurate number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Star_number
Integers occurring in the coefficients of the Taylor series of 1/cosh t
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Euler_number
Function whose domain is the positive integers
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Arithmetic_function
Repeated sum of a number's digits
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Digital_root
Number of paths between grid corners, allowing diagonal steps
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Delannoy_number
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
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
Used as a sub-base in Supyire. 90 Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2).
List_of_numeral_systems
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
Number that cannot be written as an aliquot sum
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Untouchable_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
— 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
Number that when multiplied by another number moves its last digit to its front
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Parasitic_number
Number of dots in a centred dot square
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Centered_square_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
Number in the 5th cell of any row of Pascal's triangle
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Pentatope_number
Natural number
has a prime composite index of 1031, also the index of the fifth prime repunit (of just 1s) in base-ten, R1031 (following R2, R19, R23, and R317). "1234"
1234_(number)
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
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
Methods to test or prove primality
R(109297) = ( 10 109297 − 1 ) / 9 {\displaystyle (10^{109297}-1)/9} , a repunit of 109297 digits. The certification was performed Paul Underwood using
Elliptic_curve_primality
Type of figurate number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Polygonal_number
Power of a prime number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Prime_power
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
Type of figurate number
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Hexagonal_number
Pair of integers related by their divisors
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Amicable_numbers
Size of a geometric arrangement of points
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Figurate_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
Integer divisible by sum of its digits
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Harshad_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
Polyhedral number representing a tetrahedron
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Tetrahedral_number
Numbers whose prime factors all divide the number more than once
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Powerful_number
Area of a right triangle with rational-numbered sides
Automorphic Trimorphic Digit-composition related Palindromic Pandigital Repdigit Repunit Self-descriptive Smarandache–Wellin Undulating Digit-permutation related
Congruent_number
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