257

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 $10$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$, $113$, $193$, $241$:
 * $257 = 1 \times 2^8 + 1$


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


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

Also see

 * Construction of Regular $257$-Gon