Battle of Honey Springs

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

257 is a prime number of the form 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 ➜ .[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

  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.
source: https://en.wikipedia.org/wiki/257_(number)