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The smallest number that is the sum of three cubes in two ways: 1^{3} + 5^{3} + 5^{3} = 2^{3} + 3^{3} + 6^{3}. In 1644, Marin Mersenne declared that 2^{251}  1 was composite, but gave no proof. An anonymous seventh grader once said "I noticed that 251, 521, and 2^{521}  1 are all prime. Why don't you try 512 * 2^{512}  1 or something?" 251 can be expressed using the first three primes: 2^{3} + 3^{5}. [Sladcik] The sum of the letters of "two hundred and fifty one" is the only selfdescribed prime if we use the alphabet code, i.e., a = 1, b = 2, c = 3, etc. [Lundeen] 251 is the only threedigit 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 16by16 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 taufunction 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 starts the first sequence of four consecutive primes containing the Beast number in their repeated differences: 6,6,6. Remember that the Horsemen of the Apocalypse are four. [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 (F_{251}) has a sum of digits equal to 251. The two smaller prime numbers with this property are 5 and 31. [Meller] 10^79251 is the nearest prime to the â€œUniverse number." (See, PC79). [Loungrides] Area code 251 is a telephone numbering plan code covering southwestern Alabama. [Pittman] There are 251 1/4Smith numbers below 10^9. [Gupta] (251, 257, 263, 269) is the smallest sexy prime quadruplet and the only with 3digit 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, ... .
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]
(There are 9 curios for this number that have not yet been approved by an editor.)
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