# 13

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## 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 $2 x^2 + 11$:
$2 \times 1^2 + 13 = 13$ (Previous  ... Next)

The $3$rd Proth prime after $3$, $5$:
$13 = 3 \times 2^2 + 1$

The $3$rd of $5$ primes of the form $2 x^2 + 5$:
$2 \times 2^2 + 5 = 13$ (Previous  ... Next)

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

The index of the $6$th Mersenne number after $1$, $2$, $3$, $5$, $7$ which Marin Mersenne 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 $x^2 + 1 = 2 y^4$:
$239^2 + 1 = 2 \times 13^4$

## Historical Note

The number $13$ is traditionally unlucky.

This is apparently based on the fact that there were $13$ people who attended the Last Supper.

It is believed that this superstition originated in the middle ages.

The word triskaidekaphobia means fear of the number $13$.

The fact that $13$ is in fact classified by number theorists as a lucky number is just one more indication of how mathematicians delight in confusing muggles.

Apart from that, there are several instances of the number $13$ in the mundane world:

There are $13$ times $4$ weeks in a year
There are $13$ cards in each of the $4$ suits of a deck of cards
The number of objects in a baker's dozen.