257 (number) explained

Number:	257
Prime:	yes

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

257 is a prime number of the form

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

n ^{n} + 1 ➜ 4 ^{4} + 1 = 257 .^[2]

It is alsoa 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]

Notes and References

1. .
2. Web site: Weisstein. Eric W.. Sierpiński Number of the First Kind. 2020-07-30. mathworld.wolfram.com. en.
3. Balanced primes.
4. Irregular primes.
5. Primes of form n^2 + 1.
6. Jacobsthal-Lucas numbers.
7. 2024-02-21 .
8. 2024-02-21 .
9. Number of polyhedra (or 3-connected simple planar graphs) with n nodes.