# Unique Numbers Properties

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**Unique numbers :**

Every number has its on speciality or it has some unique properties.The properties of some Unique Numbers are given below.

**1**

• 1 is the multiplicative identity

• 1 is the only positive integer that is neither prime nor composite

• 1 is its own factorial, and its own square and cube (and so on, as 1 * 1 * ... * 1 = 1)

• 1 is the only number whose concatenation with itself can yield primes in many cases

• 1 is the only number with exactly one positive divisor

• 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few

• 1 is the only square-free square

• 1 is the only triangular cube (proved by Max Alekseyev and Jaap Spies)

**2**

• 2 is the smallest prime number

• 2 is the only even prime number

• 2 is the only number whose factorial is prime

• For any polyhedron, 2 is the number of vertices plus the number of faces minus the number of edges

• The smallest field has 2 elements

• 2 is the only number that it is its own primorial, {In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied. }as well as its own factorial

• 2 is the only number that isn't n-polygonal with any n

• 2 is the largest deficient factorial

**3**

• 3 is the greatest number of consecutive integers that can be pair wise relatively prime

• 3 is the only prime sandwiched between a prime and a composite number

• 3 is the only prime followed by a square

• Every positive integer is the sum of at most 3 triangular numbers

• If the n-th Fibonacci number is prime, then n must itself be prime, with the exception of 3, which is the 4th Fibonacci number

• 3 is the only number which is equal to the sum of all the natural numbers less than it

• 3 is the only triangular number that is also prime

**4**

• 4 is the only compositorial square

• 4 is the only positive number that is both the sum and the product of the same two integers

• 4 is the order of the smallest non-cyclic group

• Every positive integer is the sum of at most 4 squares

• 4 is the smallest number of colors sufficient to color any planar map

• 4 is the only number in the English language for which the number of letters in its name is equal to the number itself

• 4 is the only composite number that is equal to the sum of its prime factors

• 4 is the only composite number n which doesn't divide (n-1)!

**5**(unique numbers)

• 5 is the smallest number of queens needed to attack every square on a standard chess board

• 5 is the only prime which is the difference of two squares of primes

• 5 is the only prime that is a member of 2 pairs of twin primes

• 5 is the only number which is equal to the sum of all primes less than itself

• 5 is the number of Platonic { In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex} solids

• 5 is the smallest degree at which polynomial roots are no longer findable in closed form

• 5 is the smallest odd prime which is not a Gaussian prime

• 5 is the smallest number of vertices needed to create a non-planar graph

• The only polygon that has the same number of sides and diagonals is a pentagon

• 5 is the only number that is a member of two different pairs of twin primes

**6**(unique numbers)

• 6 is the only even evil perfect number

• 6 is the order of the smallest non-abelian { In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers.} group

• 6 is the only number (except 1) such that the sum of all the primes up to 6 equals the sum of all the composite numbers up to 6 (inclusive)

• 6 is the only even perfect number, for which repeatedly summing the digits you do not get 1

• 6 is the only mean between a pair of twin primes which is triangular

• The symmetric group S6 is the only finite symmetric group which has an outer automorphism { In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. }

• 6 is the minimum number of colors that is always sufficient to color any map on a Klein bottle or on a Möbius strip

• 6 is the smallest perfect number

• 6 is the only number that is both the sum and the product of the same three distinct positive integers

• 6 is the only square-free perfect number

• 6 is the only even perfect pronic {A figurate number of the form , where is the th triangular number. } • number

• 6 is the largest integer to be both a factorial and a primorial

• 6 is the only perfect number that is also a product of its proper divisors

• 6 is the largest square-free factorial

• 6 is the only perfect factorial

• 6 is the number of convex regular polychora in 4D space

**7**(unique numbers)

• 7 is the smallest number that cannot be represented as a sum of 3 squares

• 7 is the smallest integer that is not the difference of two primes

• There are 7 frieze groups, infinite discrete symmetry groups for patterns on a strip (infinitely wide rectangle)

• The numbers on opposite sides of a regular die always add up to 7

• 7 is the minimum number of colors that is always sufficient to color any map on a torus

• 7 is the only prime followed by a cube

• 7 is the smallest n for which a regular polygon with n sides cannot be constructed with ruler and compass

• The Fano plane is the projective plane with the least number of points and lines:

**8**

• 8 is the only composite cube in the Fibonacci sequence

• 8 is the dimension of the octonions and is the highest possible dimension of a normed division algebra

• 8 is the smallest number (except 1) which is equal to the sum of the digits of its cube

**9**

• 9 is the smallest odd composite number

• 9 is the maximum number of cubes that are needed to sum to any positive integer

• A cat has 9 lives

• There were 9 planets in our solar system before August 2006

• There are 9 members of The Fellowship of the Ring

• 9 is the only number (except one) which is equal to the sum of the digits of its square

• 9 is the only non-trivial square consisting of only odd digits

• 9 is the smallest number of distinct integer-sided squares needed to tile a rectangle • 9 is the smallest odd number such that no odd Fibonacci number is divisible by it

**10**(unique numbers)

• 10 is the base of our number system

• In base n, n is always written "10"

• 10 is the only semi-prime number with the property that the sum as well as the difference of its prime divisors are primes (2 + 5 = 7 and 5 - 2 = 3)

• 10 is the smallest no quotient, a number that can not be expressed as the difference between any integer and the total of coprimes below it

**11**(unique numbers)

• 11 is the smallest prime such that 2p-1 is not prime

• 11 is the largest number which is not expressible as the sum of two composite numbers

• 11 is the smallest prime for which the sum of digits equals the number of digits

• 11 is the only prime comprising an even number of identical digits

• 11 is the smallest strobogrammatic{ A strobogrammatic prime is a prime number that, given a base and given a set of glyphs, appears the same whether viewed normally or upside down.} prime

• 11 is the only palindromic prime with an even number of digits

• 11 is the only prime whose period length is two

• 11 divides all palindromes with an even number of dig

its

**12**(unique numbers)

• 12 is the smallest abundant number

• 12 is the kissing number in three dimensions

• There are 12 pentominos, the polyominoes made from 5 squares

• 12 is the only number such that it is equal to the sum of 3 raised to its digits: 12 = 3

^{1}+ 3

^{2}

• 12 is the smallest number such that it is equal to the sum of its digits plus the cubes of its digits: 12 = 1 + 2 + 1

^{3}+ 2

^{3}

• 12 is the weight of the modular discriminant as a modular form

**13**

• 13 is widely known as the unlucky number (it also happens to be a Lucky number)

• 13 is the smallest prime which is not a minimal prime

• Three planes can cut a donut into a maximum of 13 parts

• 13 is the smallest prime which can be represented as sum of two primes (i.e., 2 + 11) as well as sum of two composite numbers (i.e., 9 + 4)

• 13 is the smallest emirp

• 13 is the smallest prime whose sum of digits is square

• 13 is the number of Archimedian solids

• Unreasoned fear of the number 13 is termed triskaidekaphobia

**14**

• 14 is the largest number for which there are as many composite numbers less than it as there are primes

• 14 is the lowest even n for which the equation φ(x) = n has no solution, making it the first even nontotient

• 14 is the smallest positive integer n such that n and 2n end with the same digit

**15**(unique numbers)

• 15 is the smallest emirpimes

• 15 is the smallest composite cyclic number, that is number n with the property that there is only one group of order n

• 15 is the magic constant of the unique order-3 normal magic square

• 15 is the number of letters in the words "uncopyrightable" and "dermatoglyphics", which are the only two longest words there are without repeating a letter

**16**(unique numbers)

• 16 is the number of vertices of a tesseract

• 16 is the only number of the form x

^{y }=y

^{x}with different x and y

• 16 is the smallest prime power of a prime power of a prime

• 16 is the base of the hexadecimal number system, which is used extensively in computer science

**17**

• There are 17 distinct sets of regular polygons that can be packed around a point (e.g., 4 squares, 2 hexagons and 2 triangles, etc)

• There are exactly 17 ways to express 17 as the sum of 1 or more primes - 17 is the only integer which is equal to the number of prime partitions of itself

• 17 is the number of wallpaper groups

• 17 is the only prime of the form p

^{q}+ q

^{p}, where p and q are prime

• 17 is the only multidigit number n such that n + SOD(n) and n - SOD(n) are square numbers, where SOD means sum of digits

• 17 is described at MIT as 'the most random number', according to hackers' lore

• 17 is the number of syllables in a haiku

• 17 is the smallest odd prime such that no odd Fibonacci number is divisible by it

**18**(unique numbers)

• 18 is the smallest difference between an emirp and its reverse

• 18 is the only number that is twice the sum of its digits

**19**

• 19 is the side length of the Go board

• 19 is the largest prime which is a palindrome in Roman numerals

• 19 is the smallest number n such that n

^{n}is pandigital (contains all 10 digits)

• 19 is the smallest prime which when turned upside down yields a different prime (61)

• 19 is the smallest prime with a digital root of 1

• The 19th Fibonacci number is the smallest composite Fibonacci number with a prime index

• 19 is the smallest prime that has the same digit sum as its square

• 19 is the only number n for which there is no multidigit palindrome whose digit sum and beginning are both n

• 19 is the smallest prime whose reversal is composite

• 19 is the only prime which is equal to the difference of two cubes of primes

• 19 is the smallest number that is equal to the product of its digits plus the sum of its digits

**20**(unique numbers)

• In a game of chess both players have 20 first moves from which to choose

• 20 is the smallest number that cannot be either prefixed or followed by one digit to form a prime

**21**(unique numbers)

• 21 is the smallest number of distinct integer-sided squares needed to tile a square

• 21 is the number of spots on a standard cubical die (1+2+3+4+5+6)

**22**

• 22 is the smallest Hoax Number

• 22 is the smallest multidigit number such that the sum of its digits equals the product of its digits

**23**(unique numbers)

• 23 is the smallest group of people where there is more than a 50% chance that 2 people will share the same birthday (day and month, not year)

• 23 is the smallest isolated prime, i.e., not an element of a set of twin primes

• 23 is the smallest prime whose reversal is a power: 32 = 2

^{5}

• 23 is the only prime p such that p! is p digits long

• 23! is the least pandigital factorial, that is it contains all the digits 0 through 9 at least once

• 23 is the smallest prime p such that the ring of integers in the cyclotomic field of pth roots of unity does not have unique factorization

• 23 is the smallest Pillai Prime

**24**

• 24 is the only number that is the product of all the numbers less than its square root

• 24 divides the difference between any two prime squares greater than three

• Subtracting one from each of its divisors (except 1 and 2, but including itself) yields a prime number - 24 is the largest number with this property

• 24 is the smallest abundant factorial

**25**(unique numbers)

• 25 is the smallest square that can be written as a sum of 2 squares

• 25 is the only prime square whose digits are all prime

• 25 is the smallest aspiring number

**26**

• 26 is the only natural number to be sandwiched between a square and a cube

• 26 is the smallest non-palindrome with a palindromic square

**27**(unique numbers)

• 27 is the only number which is thrice the sum of its digits

• 27 is the first composite number not divisible by any of its digits

• 27 is the largest number that is the sum of the digits of its cube

• 27 is the only 2-digit number in which the sum of digits is equal to the sum of prime factors (27 = 3 * 3 * 3 and 2 + 7 = 3 + 3 + 3 = 9)

• A 10,000-day-old person is 27 years old

• 27 is the smallest cube out of two known with only prime digits (the other cube is 3375)

• 27 is the smallest evil cube

**28**

• 28 is the number of dominoes in standard domino sets

• 28 is the only even perfect number of the form x

^{3}+ 1

**29**

• The 29th power of two is the largest power of two to have all different digits

• 29 is the smallest multidigit prime whose product of digits of its cube is also a cube

**30**(unique numbers)

• 30 is the first sphenic number, that is the smallest number with 3 distinct prime divisors

• 30 is the largest number with the property that all smaller numbers coprime to it are prime

• Both a dodecahedron and an icosahedron have 30 edges

**31**

• 31 is the earliest and the only known case such that the sum of the divisors of two distinct numbers (16 & 25) is the same prime quantity (31), that is to say: 1+2+4+8+16 = 31 and 1+5+25 = 31

• 31 is the number of minimal composites which cover the set of composites in base 10

• There are only 31 numbers which cannot be expressed as the sum of distinct squares

• 31 = 2

^{2}+ 3

^{3}, i.e., the sum of the first two primes raised to themselves.

**32**(unique numbers)

• 32 is conjectured to be the highest power of two with all prime digits

**33**

33 is the largest number that is not a sum of distinct triangular numbers.

**34**

• 34 is the smallest number with the property that it and its neighbors have the same number of divisors

• 34 is the magic constant of a 4 by 4 magic square

• 34 is the alphanumeric value of ONE

**35**

• 35 = 2,

^{3}+ 3

^{3}, i.e., the sum of the cubes of the first two primes

• There are 35 hexominos, the polyominoes made from 6 squares

**36**(unique numbers)

• 36 is the smallest number out of two (the other being 360) that have the same number of letters in its Roman representation as its double, triple, quadruple, quintuple, sextuple and septuple

• 36 is the smallest number (besides 1) which is both square and triangular

• 36 is the smallest square that is the sum of a twin prime pair: 17 and 19

• On the piano, 36 is the number of black keys

• 36 is the smallest number containing all the digits when raised to the 10th power

• 36 is the smallest number n such that n and 2

^{n}end with the same two digits

• 36 is the smallest abundant square number

• 36 is the smallest abundant triangular number

**37**

• 37 is the smallest irregular prime

• 37 is the smallest left and right truncatable prime having more than one digit

• 37 is the only prime with period length three: 1/37 = 0.027 027 027 ...

• 37 is the prime you get if a three digit number having the same digits is divided by its digit sum

**38**

• 38 is the magic constant in the only possible magic hexagon (which utilizes all the natural integers up to and including 19)

• XXXVIII (=38) is lexicographically the last string which represents a valid Roman numeral

• 38 is the largest even number which cannot be written as the sum of two odd composite numbers

**39**

• 39 is the largest number that has the same number of letters in its Roman numeral representation as its square

• 39 is the smallest number whose sum of digits is larger than that of its square

• 39 is the smallest number with multiplicative persistence 3

• 39 is the smallest number which has 3 different partitions into 3 parts with the same product

• 39 is the least composite odd number that is the sum of the primes between its smallest and largest prime factors (39 = 3 x 13 = 3 + 5 + 7 + 11 + 13)

**40**

• In English, forty is the only number whose constituent letters appear in alphabetical order

**41(unique numbers)**

• 41 is the smallest non-palindromic prime which on subtracting its reverse gives a perfect cube (i.e., 41 - 14 = 3

**43**• 43 is the smallest prime formed from reverse concatenation of two consecutive numbers • 43 is the smallest prime whose index (14) is divisible by the sum of its digits (4+3) • 43 is the smallest prime that is not the sum of two palindromes • 43 is the smallest non-palindromic prime which on subtracting its reverse gives a perfect square (i.e., 43 - 34 = 3) • 43 is the minimal sum on each face of Honaker's Magic Die - a die with dots numbered with distinct integers to sum up to the same sum on each face

^{2}

^{2}

**45**

**46**• 46 is the number of human chromosomes • 46 is the largest even integer for which there is no pair of abundant numbers that add up to it

^{n}

**48**• 48 is the smallest number with 10 divisors • 48 is the only two-digit number that equals the difference of the squares of its digits: 48 = 8 - 4

**49**• 49 is the number of strings on a harp and the number of keys on a celesta • 49 is the smallest number which is the concatenation of two prime squares creating another prime square • 49 is the largest prime square which is greater than the product of all lesser primes • 49 is the smallest number with the property that it and its neighbors are squareful

**51**

**52**• 52 is the number of cards in a deck • The month and day are simultaneously prime a total of 52 times in a non-leap year • On the piano, the number of white keys is 52

^{3}

^{7}

^{9}

^{53}

**54**• 54 is the number of colored squares on a Rubik's cube (unique numbers) • 55 is the largest triangular number in the Fibonacci sequence • 55 is the smallest multidigit triangular number which is a palindrome (unique numbers) 56 is the number of reduced 5×5 Latin squares.

**58**• 58 is the smallest Smith number with a prime sum of digits

**60**• The smallest non-abelian simple group (the alternating group on 5 elements) has order 60, in particular 60 is the smallest composite which is the order of a simple group • 60 is the smallest product of the sides of a Pythagorean triangle

^{p}

**62**• 62 is the smallest inconsummate number in base 10: no number is a 62-multiple of the sum of its digits • 62 is the only number whose cube (238328) consists of 3 digits each occurring 2 times (unique numbers) • 63 is the smallest number out of two (the other being 69) such that the common alphabetical value of its Roman representation is equal to itself (LXIII - 12+24+9+9+9 = 63)

**65**

^{2}

**66**• In Star Wars, Order 66 is a prepared order to the clone troopers to kill the Jedi commanding them • 66 is the smallest abundant palindrome (unique numbers) • 67 is the only number such that the common alphabetical value of its Roman representation is equal to its reversal (LXVII - 12+24+22+9+9=76) • 67 is the smallest prime which contains all ten digits when raised to the tenth power • 67 is the largest prime which is not the sum of distinct squares

**69**

^{69}

^{2}

^{3}

**70**• 70 is the smallest weird number • 70 is the largest known number n such that 2 has a digit sum of n (the only other such number known is 5) • 70 is the only non-trivial square number of cannonballs that can be piled into a square pyramid

^{n}

**72**

**73**• Pi Day occurs on the 73rd day of the year (March 14) on non-leap years • 73 is the largest integer with the property that all permutations of all its substrings are primes • 73 is the largest two-digit Unholey prime: such primes do not have holes in their digits • 73 is the smallest number (besides 1) which is one less than twice its reverse • 73 is the alphanumeric value of the word NUMBER: 14 + 21 + 13 + 2 + 5 + 18 = 73

**75**• 75 is the smallest pandigital number in base 4: 75 is written as 1023 in base 4 and contains all the possible digits (unique numbers) 76 is an automorphic number.

**78**• A standard Tarot deck containing the 22 cards of the major arcana and the 56 cards of the minor arcana make up 78 cards • 78 is the number of lines that make up Metatron's Cube

**80**• 80 is the smallest number that is diminished by taking its sum of letters (writing out its English name and adding the letters using a=1, b=2, c=3, ...) - EIGHTY = 5+9+7+8+20+25 = 74 (unique numbers) • 81 is the only integer (except 1) which is the square of the sum of its digits

**83**• There are exactly 83 right-truncatable primes • 83 is the smallest multidigit prime consisting of only curved digits • 83 is the smallest prime number which is the sum of a prime number of consecutive primes in a prime number of different ways (unique numbers) • 84 is the smallest number that can be expressed as the sum of 3 distinct primes raised to distinct prime exponents: 84 = 2 + 3 + 5

**85**85 is the largest n for which 1+2+3+ ... +n = 1+2+3+ ... +m has a solution. (unique numbers) • 86 is conjectured to be the largest number n such that 2 (in decimal) doesn't contain a 0

**88**

**89**

^{q}

^{p}

**90**

**91**

**92**• 92 is the number of different placements of 8 non-attacking queens on a chessboard • 92 is the number of Johnson solids • 92 is the number of faces of a snub dodecahedron, the Archimedean solid with the most faces • The Guinness record of the longest place name, Tetaumatawhakatangihangakoauaotamateaurehaeaturipukapihimaungahoronukupokaiwhenuaakitanarahu, has 92 letters

**94**

**95**95 is the number of planar partitions of 10.

**97**• 97 is the smallest prime that has a prime alphabetical value in its Roman numerals-based representation (XCVII): 24 + 3 + 22 + 9 + 9 = 67 • There are 97 leap days every 400 years in the Gregorian Calendar • 97 is the largest two-digit prime

**99**

**100**

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