13

== Number ==
$13$ (thirteen) is:
:The $6$th prime number, after $2$, $3$, $5$, $7$, $11$
:The $1$st power of $13$ after the zeroth $1$:
::$13 = 13^1$
:The $1$st emirp:
::$13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $\ldots$
:The larger of the $3$rd pair of twin primes, with $11$
:The $2$nd Pythagorean prime after $5$, and so by Fermat's Two Squares Theorem the sum of two squares uniquely:
::$13 = 4 \times 3 + 1 = 4 + 9 = 2^2 + 3^2$
:The $2$nd prime number $p$ after $3$ the period of whose reciprocal is $\dfrac {p - 1} 2$:
::$\dfrac 1 {13} = 0 \cdot 076923 \, 076923 \ldots$
:The $2$nd integer after $7$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$13^2 = 169$; $16 = 4^2, 9 = 3^2$
:The $2$nd Wilson prime after $5$:
::$13^2 \divides \paren {13 - 1}! + 1 = 479 \, 001 \, 601$
:The $2$nd after $4$ in the sequence formed by adding the squares of the first $n$ primes:
::$13 = \ds \sum_{i \mathop = 1}^2 {p_i}^2 = 2^2 + 3^2$
:The $2$nd of $11$ primes of the form $2 x^2 + 11$:
::$2 \times 1^2 + 13 = 13$ ()
:The $3$rd Proth prime after $3$, $5$:
::$13 = 3 \times 2^2 + 1$
:The $3$rd of $5$ primes of the form $2 x^2 + 5$:
::$2 \times 2^2 + 5 = 13$ ()
:The $4$th Fibonacci prime after $2$, $3$, $5$.
:The $4$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime, after $3$, $5$, $11$:
::$13 \# - 1 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 - 1 = 30 \, 029$
:The $4$th happy number after $1$, $7$, $10$:
::$13 \to 1^2 + 3^2 = 1 + 9 = 10 \to 1^2 + 0^2 = 1$
:The index of the $5$th Mersenne prime after $2$, $3$, $5$, $7$:
::$M_{13} = 2^{13} - 1 = 8191$
:The $5$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $\ldots$
:The $5$th left-truncatable prime after $2$, $3$, $5$, $7$
:The $6$th permutable prime after $2$, $3$, $5$, $7$, $11$.
:The index of the $6$th Mersenne number after $1$, $2$, $3$, $5$, $7$ which asserted to be prime
:The $7$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$:
::$13 = 5 + 8$
:The $7$th odd positive integer after $1$, $3$, $5$, $7$, $9$, $11$ that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
:The $8$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$:
::$13 = 2 + 11$
:The $9$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.
:The $11$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $11$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ such that $2^n$ contains no zero in its decimal representation:
::$2^{13} = 8192$
:The hypotenuse of the $5-12-13$ Pythagorean triangle.
:The square of the reverse of $13$ equals the reverse of the square of $13$:
::$13^2 = 169$
::$31^2 = 961$
:With $x = 239$, the only $y$ which is the solution of the indeterminate Diophantine equation $x^2 + 1 = 2 y^4$:
::$239^2 + 1 = 2 \times 13^4$
== Also see ==
* Recurring Parts of Multiples of One Thirteenth
* Twelve Factorial plus One is divisible by 13 Squared
* Square of Reversal of Small-Digit Number
* Solution of Diophantine Equation $x^2 + 1 = 2 y^4$
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== Historical Note ==
== Sources ==
* : Glossary
* : $13$
* : Glossary
* : $13$
Category:Specific Numbers
Category:13