# 257

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## Number

$257$ (two hundred and fifty-seven) is:

The $55$th prime number

The $3$rd (and largest known) prime Sierpiński number of the first kind after $2$, $5$:
$257 = 4^4 + 1$

The $4$th Fermat number and Fermat prime after $3$, $5$, $17$:
$257 = 2^{\paren {2^3} } + 1 = 2^8 + 1$

The $4$th Sierpiński number of the first kind after $2$, $5$, $28$:
$257 = 4^4 + 1$

The $6$th balanced prime after $5$, $53$, $157$, $173$, $211$:
$257 = \dfrac {251 + 263} 2$

The $7$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$:
$257 = 16^2 + 1$

The index of the $12$th and last Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $67$, $127$ which Marin Mersenne asserted to be prime
(in this case he was not correct: in $1922$, Maurice Kraitchik proved that $M_{257}$ is composite)

The $21$st long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $181$, $193$, $223$, $229$, $233$