257 (two hundred [and] 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. ^[1]

Analogously, 257 is the third Sierpinski prime of the first kind, of the form ${\displaystyle n^{n}+1}$ ➜ ${\displaystyle 4^{4}+1=257}$. ^[2]

It is also a balanced prime, ^[3] an irregular prime, ^[4] a prime that is one more than a square, ^[5] and a Jacobsthal–Lucas number. ^[6]

Four-fold 257 is 1028, which is the prime index of the fifth Mersenne prime, 8191. ^{[7] [8]}

There are exactly 257 combinatorially distinct convex polyhedra with eight vertices (or polyhedral graphs with eight nodes). ^[9]

References