257 (number)

257 is a natural number that lies between 256 and 258. It is a prime number, meaning only 1 and itself divide it evenly.

The most notable property of 257 is that it equals 2⁸ + 1, making it one of the known Fermat primes. Because of this, it plays a role in the geometry of constructible polygons (those that can be drawn using only straightedge and compass).

Mathematical properties

257 is a prime number. When prime numbers are listed starting from 2, it occupies the 55th position. It can also be expressed as the sum of two squares: 257 = 16² + 1². In binary notation, it is written as 100000001, which is a palindrome when read in either direction.

Fermat number F₃

Fermat numbers have the form F_n = 2^(2ⁿ) + 1. When n = 3, the result is 2⁸ + 1 = 257. Five known Fermat primes are known: 3, 5, 17, 257, and 65537. Pierre de Fermat conjectured that all Fermat numbers are prime, but Leonhard Euler showed that F₅ is composite, disproving the conjecture.

Geometry: the 257-gon

According to Carl Friedrich Gauss, a regular polygon with n sides can be constructed with straightedge and compass only if n is a product of a power of 2 and any number of distinct Fermat primes. Since 257 is a Fermat prime, a 257-sided regular polygon is constructible in principle, though the actual construction would be extremely lengthy.

Other uses

In computer science, 257 appears when one is added to the maximum value a byte can hold, which is 255, or when counting the total possible states of a byte including a special designated value. Additionally, 257 is a historical year in the third century of the Common Era.