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The first prime number. The ancient Greek mathematicians started counting at 2. [Lenstra] Fermat's Little Theorem tells us that if p is prime, then p divides 2^{p}  2. Every number of the form 2^{p1}(2^{p}  1), where 2^{p}  1 is prime, is an even perfect number. A "goodytwoshoes" is someone who thinks they are perfect. Consider the first 2 primes, i.e., 2 and 3. It is interesting that 2^{10} is quite close to 10^{3} in base 10. [Wells] The number of odd entries in the nth row of Pascal's triangle is 2 raised to the number of ones in the binary expansion of n. [Su] The number of representations of n as the sum of two primes is, at most, the number of primes in the interval [n/2, n  2]. [MOC] Divisibility test for 2: A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. 2! + 2 = 2^{2}. [Sladcik] If a polygon has n sides, then n  2 triangles are formed. [Glencoe] Every Fermat number is the product of all previous Fermat numbers plus 2. The smallest untouchable number, i.e., an integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). The first few are 2, 5, 52, 88, 96, .... 2^2^2^2  2^2^2 divides n^2^2^2  n^2^2 for all n. [Selfridge] Euler's formula: V  E + F = 2. For any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. () = 2. [Kulsha] The probability that the greatest prime factor of a random integer n is greater than the square root of n equals the natural logarithm of 2. [Schroeppel] 2 is the only prime digit p with prime number_of_letters (nol) in p, nol = 3, and prime sum p + nol = 5. [Seidov] Fermat's Last Theorem: The equation x^{n} + y^{n} = z^{n} has no solution in positive integers for n greater than 2. [Wiles] If 2^{p}  1 is prime then p is prime. [Murthy] The length of the hypotenuse is 2 times the shorter leg in a 306090 triangle. [Sargent] The smallest prime of the form 2^{p}  p. [Capelle] The Pythagoreans considered 2 to be the first feminine number. The Pythagoreans were embarrassed by the discovery that the square root of 2 is an irrational number, so they tried to keep this fact a secret. [Malinin] 2 is the only prime of form n^{n} + n. [Luhn] 2r represents the circumference of a circle of radius r. [Apostol] The only difference between two consecutive primes which is prime. [Luhn] The only even prime number, therefore the "oddest" prime of all. Any number with only two positive divisors is a prime number. [Gupta] Bertrand's Postulate states that there is always a prime between n and 2n. You need only one hand to slap but 2 to clap. [Murthy] (2^2^2 + 1)^(2^2^2 + 1)  (2^2^2 + 1)  1 is a 21 digit prime. [Luhn] The number of authors of the EvenMansour Cipher is 2, which is odd, because one of the authors is Even. [Croll] The only prime which is the average of two consecutive terms of the Lucas sequence. [Rupinski] The floor function of phi^phi = 2, where phi is Golden Ratio. [Gupta] UCLA mathematician and prime number researcher Terence Tao taught himself arithmetic at age 2. The first 2 primes are the only 2 primes which are minimal primes in all bases. [Rupinski] Given any even digit E and any odd digit O, integer D, and 0 < R < 2^{D}, there is exactly one number D digits in length containing only the digits E and O which leaves remainder R when divided by 2^{D}. [Rupinski] It can be shown that the probability that the greatest prime factor of a random integer n is greater than sqrt(n) is ln 2. [Rupinski] The shortest possible game of chess ending in checkmate (Fools Mate) has only 2 moves played by each side. [Patterson] 2! is the only factorial that is prime. There is no prime between n! + 2 and n! + n. [Capelle] Pseudoprimes to base 2 are sometimes called Poulet numbers. !2 + 1 is prime. Note that !2 represents subfactorial 2. [Gupta] The smallest field in abstract algebra has 2 elements. 2^2  1 is the first Mersenne prime. [Rajh] The smallest dihedral prime. [Patterson] The only prime p such that p times reversal(p)  1 is prime. [Firoozbakht] "When you're one step ahead of the crowd you're a genius. When you're 2 steps ahead, you're a crackpot." Rabbi Shlomo Riskin 2 is the smallest Kynea prime. [McAlee] Is it true that only 2 books of the King James Version of the Bible end in a question mark? Yes, the books of Jonah and Nahum. The first prime Bell number. Such numbers represent the number of ways a set of n elements can be partitioned into nonempty subsets. Named after Eric Temple Bell, a prolific Caltech math professor. [Post] 2 is the smallest prime Motzkin number. [Post] The addition and product of 2 with itself are equal, which gives it a unique arithmetic property among the positive integers. Mars is the only known planet to have two natural satellites. [Brower] De Polignac's Conjecture states that every even number is the difference of 2 consecutive primes in infinitely many ways. (n) is greater than or equal to (n/2)3/2 for each natural number n. Note that 2 and 3 are the first prime numbers. [Capelle] The number of words in the shortest verse (by number of letters) in the King James Version of the Bible (John 11:35) is "Jesus wept." [Doyle] The limiting sum in the infinite series of the reciprocal of triangular numbers. [Beedassy] The only "eban" prime, i.e., devoid of the letter 'e' in its English name. [Beedassy] A knight in the corner of the chessboard has only 2 possible moves. [Silva] The only prime that is common divisor to all odd primes gaps. [Silva] If p and q are consecutive prime numbers, then floor(p/q + q/p) = 2. Note that lim(p/q + q/p) = 2, as p and q approach infinity. [Capelle] The only prime whose cube is the sum of it s two consecutive primes. [Silva] The phase rule states that the number of degrees of freedom in a material system at equilibrium is equal to the number of components minus the number of phases plus the constant 2. (2^{}) = (^{2}). [Capelle] The only known number n such that the sum of the proper divisors of n is equal to phi(n). [Firoozbakht] Logicians Bertrand Russell (18721970) and Alfred Whitehead (18611947) went to great lengths to prove that 1 + 1 = 2, in their epochal Principia Mathematica. F. Viete (15401603) expressed as an infinite product containing only 2 (and its reciprocal 1/2). [Caldwell] It is possible to measure all of the integer distances from one to six on a sixinch ruler with just 2 marks. For example, the distance from the 2 to the right end is four inches. [Caldwell] The only prime digit whose complement is a nonprime digit. [Beedassy] The only digit d that appears exactly d times in ddigit primes. [Silva] The smallest number and only prime such that prime(n)=sigma(n). [Gupta] The smallest prime with a prime number of partitions. [Pol] "Check this out. The second derivative of e^{x} is e^{x}, right? And e^{x} evaluated at 0 is equal to 1, right? Therefore 2 has got to be a prime number." (from the paper A Curious Way to Test for Primes by Dennis P. Walsh, 2007) The Solitude of Prime Numbers by Paolo Giordano is a quiet book about 2 damaged people, Alice and Mattia. The only prime number that is not the difference of two squares. [Green] Only prime whose sum of divisors is odd. [Gupta] Only prime whose sum of divisors is prime. [Firoozbakht] 5^{3/7} is close to 2. [Wesolowski] The novel "Curios" by Richard Marsh is about some strange adventures of 2 bachelors. Anyone with an IQ in the top 2 percent of the population can join Mensa. The first of only four allNiven numbers that are primes. [Loungrides] The sum of the reciprocals of the divisors of a perfect number (including the reciprocal of the number itself) is always equal to 2. [Rupinski] There are about 2 lunar eclipses per year. [NASA] Pollen grains are tiny (only 2 cells). [Nowicki] There exists a periodic curve based on prime numbers intersected by only two curves.
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