# 2

## Number

$2$ (**two**) is:

- The $1$st prime number

- The only even prime number

### Zeroth Term

- The $0$th Euclid number:
- $2 = p_0\# + 1 = 1 + 1$

- The $0$th Thabit number, and $1$st Thabit prime:
- $2 = 3 \times 2^0 - 1$

### $1$st Term

- The $1$st (strictly) positive even number

- The $1$st (trivial, $1$-digit) palindromic prime

- The $1$st Sophie Germain prime:
- $2 \times 2 + 1 = 5$, which is prime

- The index of the $1$st repunit prime:
- $R_2 = 11$

- The index of the $1$st Mersenne prime:
- $M_2 = 2^2 - 1 = 3$

- The index of the $1$st Woodall prime:
- $2 \times 2^2 - 1 = 7$

- The $1$st second pentagonal number:
- $2 = \dfrac {1 \paren {3 \times 1 + 1} } 2$

- The $1$st of the $4$ known primes $p$ such that $\dfrac {p^p - 1} {p - 1}$ is itself prime:
- $\dfrac {2^2 - 1} {2 - 1} = 3$

- The $1$st power of $2$ after the zeroth $1$:
- $2^1 = 2$

- The $1$st Fibonacci prime

- The $1$st Lucas prime

- The $1$st permutable prime

- The $1$st untouchable number

- The $1$st primorial:
- $2 = p_1 \# = 2 \# := \ds \prod_{k \mathop = 1}^1 p_k = 2$

- The $1$st central binomial coefficient:
- $2 = \dbinom {2 \times 1} 1 := \dfrac {2!} {\paren {1!}^2}$

- The $1$st prime number of the form $n! + 1$ for integer $n$:
- $1! + 1 = 1 + 1 = 2$

- where $n!$ denotes $n$ factorial

- The $1$st prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
- $2 \# + 1 = 2 + 1 = 3$

- The $1$st of the lucky numbers of Euler:
- $n^2 + n + 2$ is prime for $n = 0$

- The $1$st term of Göbel's sequence after the $0$th term $1$:
- $2 = \paren {1 + 1^2} / 1$

- The $1$st term of the $3$-Göbel sequence after the $0$th term $1$:
- $2 = \paren {1 + 1^3} / 1$

- The $1$st Sierpiński number of the first kind:
- $2 = 1^1 + 1$

- The $1$st prime Sierpiński number of the first kind:
- $2 = 1^1 + 1$

- The $1$st even number which cannot be expressed as the sum of $2$ composite odd numbers

- The $1$st prime number which divides the sum of all smaller primes:
- $0 = 0 \times 2$

- (there are no primes smaller than $2$)

- The $1$st even integer that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes

- The $1$st positive integer which is not the sum of $1$ or more distinct squares

- The $1$st positive integer which cannot be expressed as the sum of distinct pentagonal numbers

- The $1$st prime number of the form $n^2 + 1$:
- $2 = 1^2 + 1$

- The $1$st of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
**deux**

- The $1$st integer at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied

- The $1$st primorial which can be expressed as the product of consecutive integers:
- $2 \# = 2 = 1 \times 2$

- The $1$st of the $2$nd pair of consecutive integers whose product is a primorial:
- $2 \times 3 = 6 = 3 \#$

- The $1$st tri-automorphic number:
- $2^2 \times 3 = 1 \mathbf 2$

- The $1$st Euclid prime:
- $2 = p_0\# + 1 = 1 + 1$

- The $1$st (trivially) two-sided prime

- The $1$st prime number consisting (trivially) of a string of consecutive ascending digits

- The $1$st Hardy-Ramanujan number: the smallest positive integer which can be expressed as the sum of $2$ cubes in (trivially) $1$ way:
- $2 = \map {\operatorname {Ta} } 1 = 1^3 + 1^3$

- The $1$st of $6$ integers which cannot be expressed as the sum of distinct triangular numbers

### $2$nd Term

- The $2$nd Catalan number after $(1)$, $1$:
- $2 = \dfrac 1 {2 + 1} \dbinom {2 \times 2} 2 = \dfrac 1 3 \times 6$

- The $2$nd Ulam number after $1$

- The $2$nd positive integer after $1$ such that all smaller positive integers coprime to it are prime

- The $2$nd generalized pentagonal number after $1$:
- $2 = \dfrac {1 \paren {3 \times 1 + 1} } 2$

- The $2$nd (strictly) positive integer after $1$ which cannot be expressed as the sum of exactly $5$ non-zero squares

- The $2$nd highly composite number after $1$:
- $\map {\sigma_0} 2 = 2$

- The $2$nd special highly composite number after $1$

- The $2$nd highly abundant number after $1$:
- $\map {\sigma_1} 2 = 3$

- The $2$nd superabundant number after $1$:
- $\dfrac {\map {\sigma_1} 2} 2 = \dfrac 3 2 = 1 \cdotp 5$

- The $2$nd almost perfect number after $1$:
- $\map {\sigma_1} 2 = 3 = 4 - 1$

- The $2$nd factorial after $1$:
- $2 = 2! = 2 \times 1$

- The $2$nd superfactorial after $1$:
- $2 = 2\$ = 2! \times 1!$

- The $2$nd factorion base $10$ after $1$:
- $2 = 2!$

- The $2$nd of the trivial $1$-digit pluperfect digital invariants after $1$:
- $2^1 = 2$

- The $2$nd of the (trivial $1$-digit) Zuckerman numbers after $1$:
- $2 = 1 \times 2$

- The $2$nd of the $5$ known powers of $2$ whose digits are also all powers of $2$:
- $1$, $2$, $\ldots$

- The $2$nd after $1$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

- The $2$nd of the (trivial $1$-digit) harshad numbers after $1$:
- $2 = 1 \times 2$

- The $2$nd Bell number after $(1)$, $1$

- The $2$nd positive integer after $1$ whose cube is palindromic (in this case trivially):
- $2^3 = 8$

- The $2$nd of the $1$st pair of consecutive integers whose product is a primorial:
- $1 \times 2 = 2 = 2 \#$

- The index of the $2$nd Mersenne number after $1$ which Marin Mersenne asserted to be prime

- The $2$nd (strictly) positive integer after $1$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

- The $2$nd after $1$ of $6$ integers $n$ such that the alternating group $A_n$ is ambivalent

- The $2$nd after $1$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways

- The $2$nd after $0$ of the $6$ integers which are the middle term of a sequence of $5$ consecutive integers whose cubes add up to a square
- $0^3 + 1^3 + 2^3 + 3^3 + 4^3 = 100 = 10^2$

### $3$rd Term

- The $3$rd integer $m$ after $0$, $1$ such that $m! + 1$ (its factorial plus $1$) is prime:
- $2! + 1 = 2 + 1 = 3$

- The $3$rd integer after $0$, $1$ such that its double factorial plus $1$ is prime:
- $2!! + 1 = 3$

- The $3$rd integer after $0$, $1$ which is (trivially) the sum of the increasing powers of its digits taken in order:
- $2^1 = 2$

- The $3$rd subfactorial after $0$, $1$:
- $2 = 3! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} }$

- The $3$rd integer $n$ after $-1$, $0$ such that $\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 = m^2$ for integer $m$:
- $\dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3 = 2^2$

- The $3$rd integer $m$ after $0$, $1$ such that $m^2 = \dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3$ for integer $n$:
- $2^2 = \dbinom 2 0 + \dbinom 2 1 + \dbinom 2 2 + \dbinom 2 3$

- The $3$rd palindromic integer after $0$, $1$ which is the index of a palindromic triangular number
- $T_2 = 3$

- The $3$rd palindromic integer after $0$, $1$ whose square is also palindromic integer
- $2^2 = 4$

- The $3$rd integer $n$ after $0$, $1$ such that $2^n$ contains no zero in its decimal representation:
- $2^2 = 4$

- The $3$rd integer $n$ after $0$, $1$ such that $5^n$ contains no zero in its decimal representation:
- $5^2 = 25$

- The $3$rd integer $n$ after $0$, $1$ such that both $2^n$ and $5^n$ have no zeroes:
- $2^2 = 4, 5^2 = 25$

- The $3$rd integer after $0$, $1$ which is palindromic in both decimal and ternary:
- $2_{10} = 2_3$

- The $3$rd Fibonacci number after $1$, $1$:
- $2 = 1 + 1$

### Miscellaneous

- The number of pairs of twin primes less than $10$:
- $\tuple {3, 5}$, $\tuple {5, 7}$

### Arithmetic Functions on $2$

\(\ds \map {\sigma_0} { 2 }\) | \(=\) | \(\ds 2\) | $\sigma_0$ of $2$ | |||||||||||

\(\ds \map \phi { 2 }\) | \(=\) | \(\ds 1\) | $\phi$ of $2$ | |||||||||||

\(\ds \map {\sigma_1} { 2 }\) | \(=\) | \(\ds 3\) | $\sigma_1$ of $2$ |

## Also see

### Previous in sequence: $0$

*Previous ... Next*: Sum of Cubes of 5 Consecutive Integers which is Square*Previous ... Next*: Sum of 4 Consecutive Binomial Coefficients forming Square

### Previous in sequence: $1$

#### Next in sequence: $3$

*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Sequence of Integers whose Factorial plus 1 is Prime*Previous ... Next*: Fibonacci Number*Previous ... Next*: Ulam Number*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation*Previous ... Next*: Powers of 2 and 5 without Zeroes*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares*Previous ... Next*: Integers such that all Coprime and Less are Prime*Previous ... Next*: Göbel's Sequence*Previous ... Next*: Numbers which are Sum of Increasing Powers of Digits*Previous ... Next*: Pluperfect Digital Invariant*Previous ... Next*: Consecutive Integers whose Product is Primorial*Previous ... Next*: Zuckerman Number*Previous ... Next*: Harshad Number*Previous ... Next*: Palindromic Indices of Palindromic Triangular Numbers*Previous ... Next*: Square of Small-Digit Palindromic Number is Palindromic*Previous ... Next*: Mersenne Prime/Historical Note*Previous ... Next*: Integers whose Number of Representations as Sum of Two Primes is Maximum*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1

#### Next in sequence: $4$

*Previous ... Next*: Highly Composite Number*Previous ... Next*: Superabundant Number*Previous ... Next*: Sequence of Powers of 2*Previous ... Next*: Almost Perfect Number*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Palindromes in Base 10 and Base 3*Previous ... Next*: Powers of 2 whose Digits are Powers of 2

#### Next in sequence: $5$

*Previous ... Next*: Generalized Pentagonal Number*Previous ... Next*: Catalan Number*Previous ... Next*: 3-Göbel Sequence*Previous ... Next*: Bell Number*Previous ... Next*: Number of Free Polyominoes*Previous ... Next*: Alternating Groups that are Ambivalent

#### Next in sequence: $6$

#### Next in sequence: $7$

#### Next in sequence: $8$ and above

#### Next in sequence: $9$ and above

*Previous ... Next*: Subfactorial*Previous ... Next*: Superfactorial*Previous ... Next*: Factorions Base 10*Previous ... Next*: Prime Values of Double Factorial plus 1

### Next in sequence: $1$

### Next in sequence: $3$

*Next*: Prime Number*Next*: Palindromic Prime*Next*: Sophie Germain Prime*Next*: Fibonacci Prime*Next*: Lucas Prime*Next*: Euler Lucky Number*Next*: Index of Mersenne Prime*Next*: Permutable Prime*Next*: Woodall Prime*Next*: Euclid Number*Next*: Euclid Prime*Next*: Two-Sided Prime*Next*: Index of Mersenne Prime

*Next*: Prime Numbers of form Factorial Plus 1*Next*: Prime to Own Power minus 1 over Prime minus 1 being Prime*Next*: Sequence of Prime Primorial plus 1*Next*: Numbers not Sum of Distinct Squares*Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Next*: Consecutive Integers whose Product is Primorial*Next*: Prime Numbers Composed of Strings of Consecutive Ascending Digits

### Next in sequence: $4$

*Next*: Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers*Next*: Even Integers not Sum of 2 Twin Primes

### Next in sequence: $5$

*Next*: Untouchable Number*Next*: Thabit Number*Next*: Thabit Prime*Next*: Sierpiński Number of the First Kind*Next*: Tri-Automorphic Number*Next*: Prime Sierpiński Numbers of the First Kind*Next*: Letters of Names of Numbers in Alphabetical Order/French*Next*: Prime Numbers which Divide Sum of All Lesser Primes*Next*: Prime Number Race between 4n+1 and 4n-1*Next*: Primes of Form n^2 + 1*Next*: Integers not Sum of Distinct Triangular Numbers

### Next in sequence: $6$

### Next in sequence: $7$

### Next in sequence: $8$

*Next*: Square Formed from Sum of 4 Consecutive Binomial Coefficients*Next*: Number of Twin Primes less than Powers of 10

### Next in sequence: $9$ and above

## Historical Note

The number **$2$ (two)** is understood to have been treated as a special case of a number from the earliest historical records.

Many early languages have specific forms of nouns for when two of an object are under consideration, as well as different forms for singular and plural.

To the ancient Greeks, in addition to having problems with the idea that $1$ is a number, it was questionable whether or not **two ($2$)** was actually a number either:

- While it has a beginning and an end, it has no middle
- Multiplication by $2$ consists merely of adding a number to itself, and multiplication was expected to do more than just add.

Thus $2$ was an exceptional case.

To the Pythagoreans, odd and even numbers were considered to be either male or female, but sources differ on which was which.

Some suggest that $2$ was considered to be the first male number, and said to personify the principle of **diversity**.

Such sources state that in contrast, the odd numbers were considered to be female.

However, other sources suggest that it was the even numbers which were female, while the odd numbers were male.

The dyad, as such, is never specifically defined in Euclid's *The Elements*, but introduced without definition in Power of Two is Even-Times Even Only.

In the words of Euclid:

*Each of the numbers which are continually doubled beginning from a dyad is even-times even only.*

(*The Elements*: Book $\text{IX}$: Proposition $32$)

## Linguistic Note

There are many words in natural language which indicate a twofold collection:

**dual**,**duel**,**brace**,**couple**,**pair**,**double**and**twin**

for example.

Words derived from or associated with the number $2$ include:

**bicycle**: a vehicle with $2$ wheels

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$

- Highly Abundant Numbers/Examples
- Fibonacci Numbers/Examples
- Ulam Numbers/Examples
- Göbel's Sequence/Examples
- Pluperfect Digital Invariants/Examples
- Zuckerman Numbers/Examples
- Harshad Numbers/Examples
- Mersenne's Assertion/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Highly Composite Numbers/Examples
- Superabundant Numbers/Examples
- Powers of 2/Examples
- Almost Perfect Numbers/Examples
- Generalized Pentagonal Numbers/Examples
- Catalan Numbers/Examples
- Bell Numbers/Examples
- Special Highly Composite Numbers/Examples
- Factorials/Examples
- Subfactorials/Examples
- Superfactorials/Examples
- Factorions/Examples
- Lucas Numbers/Examples
- Prime Numbers/Examples
- Palindromic Primes/Examples
- Sophie Germain Primes/Examples
- Fibonacci Primes/Examples
- Lucas Primes/Examples
- Euler Lucky Numbers/Examples
- Indices of Mersenne Primes/Examples
- Permutable Primes/Examples
- Woodall Primes/Examples
- Euclid Numbers/Examples
- Euclid Primes/Examples
- Two-Sided Primes/Examples
- Untouchable Numbers/Examples
- Thabit Numbers/Examples
- Thabit Primes/Examples
- Sierpiński Numbers of the First Kind/Examples
- Tri-Automorphic Numbers/Examples
- Prime Number Races/Examples
- Primorials/Examples
- Central Binomial Coefficients/Examples
- Second Pentagonal Numbers/Examples
- Repunit Primes/Examples
- Hardy-Ramanujan Numbers/Examples
- Specific Numbers
- 2