The smallest number that is the sum of three cubes in two ways: 13 + 53 + 53 = 23 + 33 + 63.
In 1644, Marin Mersenne declared that 2251 - 1 was composite, but gave no proof.
An anonymous seventh grader once said "I noticed that 251, 521, and 2521 - 1 are all prime. Why don't you try 512 * 2512 - 1 or something?"
251 can be expressed using the first three primes: 23 + 35. [Sladcik]
The sum of the letters of "two hundred and fifty one" is the only self-described prime if we use the alphabet code, i.e., a = 1, b = 2, c = 3, etc. [Lundeen]
251 is the only three-digit number such that 2*251^251+1 is prime. [Opao]
The Moon not only changes phases from lunation to lunation but changes in size as the Moon moves closer to and farther away from the Earth. The length of 18 fumocies is almost exactly equal to the length of 251 lunations. [Engel]
The sum of the squares of the first three pentatopic numbers (1, 5, 15). [Silva]
Members of the Vermont 251 Club attempt to visit 251 cities and towns in the state of Vermont.
Smallest odd prime in the decimal expansion of Pogson's ratio 100^(1/5), appearing in the definition of the astronomical magnitude (brightness) scale. [Post]
Area code 251 is a telephone numbering plan code covering southwestern Alabama. [Pittman]
(251, 257, 263, 269) is the smallest sexy prime quadruplet and the only with 3-digit terms), for which there are no other primes between the first and last terms except from its other two terms. [Loungrides]
The smallest positive number for p1^3 + p2^3 - p3^3 when p1, p2, and p3 are three consecutive primes (13,17,19). [Gaydos]
(There are 10 curios for this number that have not yet been approved by an editor.)