251

This number is a prime.

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 smallest of four consecutive primes in arithmetic progression (251, 257, 263, and 269). [Gallardo]

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]

All the prime numbers up to 251, without repetition, appear in Benjamin Franklin's original 16-by-16 square. [Capelle]

The sum of the squares of the first three pentatopic numbers (1, 5, 15). [Silva]

Its square is the smallest number n for which the Ramanujan tau-function tau(n) is prime. It was found by D. H. Lehmer in 1965. [Capelle]

Members of the Vermont 251 Club attempt to visit 251 cities and towns in the state of Vermont.

Michael Jackson acquired Sony/ATV Music Publishing which owned the copyrights to 251 Beatles songs.

251 is the initial prime number in the first set of four consecutive primes with common difference of six: {2516️⃣2576️⃣2636️⃣269}. Recall the Four Horsemen of the Apocalypse? [Silva]

The smallest prime which becomes a semiprime if any digit is deleted. [Post]

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]

The 251st Fibonacci number (F₂₅₁) has a sum of digits equal to 251. The two smaller prime numbers with this property are 5 and 31. [Meller]

10^79-251 is the nearest prime to the “Universe number." (See, PC79). [Loungrides]

Area code 251 is a telephone numbering plan code covering southwestern Alabama. [Pittman]

251 is the smallest prime number that can be added to a 5-digit number with unique digits to produce a 5-digit number made of the other 5 digits: 39875 + 251 = 40126. [Gaydos]

The smallest non-palindromic prime p whose cube ends with the same p digits (15813251). [Bajpai]

There are 251 1/4-Smith numbers below 10^9. [Gupta]

(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]

Puzzle Sequence: 7, 19, 79, 491, 2423, ... . Coauthor of Prime Curios!, G. L. Honaker, Jr., estimates an IQ of 251 or greater is required to determine the next term. Email him for answer and explanation.

The IQ251 is a large capacity controller designed for the control of all types of building plant.

The smallest positive number for p1^3 + p2^3 - p3^3 when p1, p2, and p3 are three consecutive primes (13,17,19). [Gaydos]

Sum of all prime number from 31 to 13 and back to 31. [Wiszowaty]