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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).

References

1. Hsiung, C. Y. (1995), Elementary Theory of Numbers, Allied Publishers, pp. 39–40, ISBN 9788170234647.
2. Weisstein, Eric W. "Sierpiński Number of the First Kind". mathworld.wolfram.com. Retrieved 2020-07-30.
3. Sloane, N. J. A. (ed.). "Sequence A006562 (Balanced primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. Sloane, N. J. A. (ed.). "Sequence A000928 (Irregular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. Sloane, N. J. A. (ed.). "Sequence A002496 (Primes of form n^2 + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. Sloane, N. J. A. (ed.). "Sequence A014551 (Jacobsthal-Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-21.
8. Sloane, N. J. A. (ed.). "Sequence A000668 (Mersenne primes (primes of the form 2^n - 1).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-21.
9. Sloane, N. J. A. (ed.). "Sequence A000944 (Number of polyhedra (or 3-connected simple planar graphs) with n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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