# 257

The largest Fermat prime *F*_{n} =
2^{2n} + 1 with n digits. [Beedassy]

The largest known prime of the form *n*^{n} + 1. It is very likely that 2 and 5 are the only others.

The smallest three-digit prime with distinct prime digits. [Moore]

257 = 2^{8} + 1, (2^{8} + 1)^{2} - 2 and (((2^{8} + 1)^{2} - 2)^{2} - 2) are prime. [Luhn]

257 is the fourth of the five known Fermat primes; the other four are 3, 5, 17, and 65537. Many mathematicians believe that no more Fermat primes exist. [Dobb]

The smallest odd octavan prime, i.e., of the form *p* = *x*^8 + *y*^8. [Russo]

Using the primes up to 257 in a sieve of Eratosthenes on the set of integers leaves only 10% of numbers unfactored. [Rupinski]

More than 90% of all positive integers are composite numbers that have a lowest prime factor of 257 or less. [Schuler]

The smallest prime with the following property: It is the decimal value of either "FIFTY PLUS EIGHTYEIGHT" or "FIFTYEIGHT PLUS EIGHTY" (with A=1, B=2, ..., Z=26), both giving "ONEHUNDREDTHIRTYEIGHT". Summing up the character representations of that string again results in 257! [Vago]

*Lab 257*, a book by Michael C. Carroll, is (the subtitle asserts) "the
disturbing story of the government's secret Plum Island germ laboratory."

The smallest prime of the form 128*k* + 1. Note that any
prime factor of *F*_{n} (where *n* is
greater than 2) is of this form.

The largest prime in a sequence of fifteen primes of the form 2*t* + 17, where *t* runs through the first fifteen triangular numbers, i.e., positive integers of the form *n*(*n* + 1)/2. [Silva]

The only prime of the form a^a + b^b where a and b are one-digit non-prime integers. [Loungrides]

The largest prime consisting only of prime digits and the absolute difference between any two of its digits is also prime. [Green]

257 = prime(|2 - 57|). It is the only Fermat prime with this property. [Firoozbakht]

According to eyewitness accounts, as many as 257 Texian may have died at the Battle of the Alamo.

The only known Fermat prime that is not a deletable prime. [Wesolowski]

The only prime-digit prime whose absolute value of the differences between any two of its digits are also prime. [Loungrides]

The smallest prime-digit prime with distinct digits that is the sum of three prime-digit primes, i.e., (7+23+227). Note that the concatenation in order of these primes, i.e., 723227 is also prime. [Loungrides]

During his imprisonment in the Temple before being guillotined, King of France Louis XVI read 257 books. Reference: Nougaret PJB (1797) Histoire des Prisons de Paris et des Départements. Paris: Courcier, vol. 1, p. 139. [Olry]

The only known Fermat prime that is *not* a member of
a twin prime pair. [Honaker]

257^100 is the smallest 100th power whose reversal is prime. [Gaydos]

*Wildflower* (a 2017 Philippine revenge drama television
series) completed with a total of 257 episodes. [Cuenta]

257 is the largest known prime of the form N^N+1. Note that if N^N+1 (for N>1) is a prime then it must be of the form 2^2^(2^m+m)+1 (m>=0). So according to the Fermat number factoring status, the next prime of the form N^N+1 (if it exists) is may be 2^2^70+1 and will have more than 8*10^20 digits (!). [Antonious]

The 257th, 258th, and 259th centuries are the smallest three consecutive centuries to have no years that are powers of other numbers, i.e., no squares, cubes, 4th powers, etc. [Gaydos]

In the Swiss card game Jass, the total number of points is 257 if one team takes all tricks in a game. [Perrenoud]

1068349 is the smallest positive integer that together with its smallest prime factor (257) is pandigital. [Gaydos]