# 17

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## Number

$17$ (seventeen) is:

The $7$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$

The $1$st integer to be the sum of $2$ distinct powers of $4$:
$17 = 1 + 16 = 1^4 + 2^4$

The $1$st odd positive integer not expressible in the form $2 n^2 + p$ where $p$ is prime

The $2$nd emirp after $13$

The $2$nd long period prime after $7$:
$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$

The $3$rd Stern number after $1$, $3$

The $3$rd Stern prime after $1$, $3$

The $3$rd Fermat number and Fermat prime after $3$, $5$:
$17 = 2^{2^2} + 1$

The $3$rd positive integer $n$ after $5$, $11$ such that no factorial of an integer can end with $n$ zeroes

The $3$rd prime number of the form $n^2 + 1$ after $2$, $5$:
$17 = 4^2 + 1$

The $3$rd integer after $2$, $5$ at which the prime number race between primes of the form $4 n + 1$ and $4 n - 1$ are tied

The $4$th Proth prime after $3$, $5$, $13$:
$17 = 1 \times 2^4 + 1$

The $4$th Dudeney number after $0$, $1$, $8$, and only prime Dudeney number:
$17 = 4 + 9 + 1 + 3$, while $17^3 = 4913$

The smaller of the $4$th pair of twin primes, with $19$

The $5$th of the lucky numbers of Euler after $2$, $3$, $5$, $11$:
$n^2 + n + 17$ is prime for $0 \le n < 15$

The index of the $6$th Mersenne prime after $2$, $3$, $5$, $7$, $13$:
$M_{17} = 2^{17} - 1 = 131 \, 071$

The $6$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$

The $7$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$

The $7$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $\ldots$

The index of the $7$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$ which Marin Mersenne asserted to be prime

The $9$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $\ldots$

The $12$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes

The $12$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$ such that $5^n$ contains no zero in its decimal representation:
$5^{17} = 762 \, 939 \, 453 \, 125$

There are $17$ wallpaper groups

## Historical Note

$17$ was the highest number whose square root was proved irrational by Theodorus of Cyrene.

Plutarch reports that the Pythagoreans had a horror of the number $17$.

That was because it lay half way between $16$ and $18$, which are the areas of rectangles whose areas equal their perimeters.

$17$ seems to be a popular age to write a pop song about:

At Seventeen by Janis Ian
7-Teen by The Regents
Edge of Seventeen by Stevie Nicks

to name but three.