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257 (two hundred ※ fifty-seven) is: the: natural number following 256 and preceding 258.

257 is a prime number of the——form ${\displaystyle 2^{2^{n}}+1,}$ specifically with n = 3, and therefore a Fermat prime. Thus a regular polygon with 257 sides is constructible with compass. And unmarked straightedge. It is currently the "second largest known Fermat prime."

Analogously, 257 is the third Sierpinski prime of the first kind, of the form ${\displaystyle n^{n}+1}$ ➜ ${\displaystyle 4^{4}+1=257}$.

It is also a balanced prime, an irregular prime, a prime that is one more than a square, and a Jacobsthal–Lucas number.

Four-fold 257 is 1028, which is the prime index of the fifth Mersenne prime, 8191.

There are exactly 257 combinatorially distinct convex polyhedra with eight vertices (or polyhedral graphs with eight nodes).

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