13

Number
$13$ (thirteen) is:


 * The $6$th prime number, after $2, 3, 5, 7, 11$


 * The $2$nd of the $3$rd pair of twin primes, with $11$


 * The second 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 $5$th lucky number:
 * $1, 3, 7, 9, 13, \ldots$


 * The $7$th Fibonacci number, after $1, 1, 2, 3, 5, 8$:
 * $13 = 5 + 8$


 * The $4$th Fibonacci prime after $2, 3, 5$.


 * The $8$th Ulam number after $1, 2, 3, 4, 6, 8, 11$:
 * $13 = 2 + 11$


 * 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 $2$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 smallest emirp:
 * $13, 17, 31, 37, 71, 73, 79, 97, 107, 113, \ldots$


 * The $6$th permutable prime after $2, 3, 5, 7, 11$.


 * The hypotenuse of the $5-12-13$ Pythagorean triangle.


 * The square of the reverse of $13$ equals the reverse of the square of $13$:
 * $13^2 = 169$
 * $31^2 = 961$


 * The $4$th of the sequence of $n$ such that $p_n \# - 1$, where $p_n \#$ denotes primorial of $n$, is prime, after $3, 5, 11$:
 * $p_{13} \# - 1 = 30 \, 029$


 * The $2$nd of $11$ primes of the form $2 x^2 + 11$:
 * $2 \times 1^2 + 13 = 13$


 * The $3$rd of $5$ primes of the form $2 x^2 + 5$:
 * $2 \times 2^2 + 5 = 13$


 * 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 $9$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 integer $n$ after $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ such that $2^n$ contains no zero in its decimal representation:
 * $2^{13} = 8192$


 * 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 $x^2 + 1 = 2 y^4$