The first prime number.
The ancient Greek mathematicians started counting at 2. [Lenstra]
Every number of the form 2p-1(2p - 1), where 2p - 1 is prime, is an even perfect number.
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]
2! + 2 = 22. [Sladcik]
If a polygon has n sides, then n - 2 triangles are formed. [Glencoe]
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]
2 is the only prime digit p with prime number_of_letters (nol) in p, nol = 3, and prime sum p + nol = 5. [Seidov]
If 2p - 1 is prime then p is prime. [Murthy]
The length of the hypotenuse is 2 times the shorter leg in a 30-60-90 triangle. [Sargent]
The smallest prime of the form 2p - p. [Capelle]
The Pythagoreans considered 2 to be the first feminine number.
2 is the only prime of form nn + n. [Luhn]
2r represents the circumference of a circle of radius r. [Apostol]
The only difference between two consecutive primes which is prime. [Luhn]
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 Even-Mansour Cipher is 2, which is odd, because one of the authors is Even. [Croll]
UCLA mathematician and prime number researcher Terence Tao taught himself arithmetic at age 2. When his parents asked him how he had learnt these skills, he replied that he had been watching Sesame Street on television.
Given any even digit E and any odd digit O, integer D, and 0 < R < 2D, there is exactly one number D digits in length containing only the digits E and O which leaves remainder R when divided by 2D. [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 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]
A knight in the corner of the chessboard has only 2 possible moves. [Silva]
The only prime whose cube is the sum of it s two consecutive primes. [Silva]
(2) = (2). [Capelle]
F. Viete (1540-1603) 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 six-inch 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 d-digit primes. [Silva]
The smallest prime with a prime number of partitions. [Pol]
"Check this out. The second derivative of ex is ex, right? And ex 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 only prime number that is not the difference of two squares. [Green]
Only prime whose sum of divisors is prime. [Firoozbakht]
53/7 is close to 2. [Wesolowski]
Anyone with an IQ in the top 2 percent of the population can join Mensa.
The first of only four all-Niven 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]
Pollen grains are tiny (only 2 cells). [Nowicki]
There exists a periodic curve based on prime numbers intersected by only two curves.
Two wrongs do not make a right. [Gupta]
The only known prime x such that 2^x+3^x, 2^x+5^x, 2^x+7^x are all primes, i.e., 13, 29, 53. [Loungrides]
Good Friday occurs 2 days before Easter Sunday in the United States.
A family room of the Prime City Resort Hotel in Angeles City, Philippines, contains 2 "twin size" beds.
(2^3*3^2) +/- 1 and (2^3*3^2)^2 +/- 1 both form a twin-prime pair. [Galliani]
168 is the largest power of the prime 2 whose decimal expansion is free of 2: 2^168 = 374144419156711147060143317175368453031918731001856. Found by Phil Carmody. [Rivera]
(There are 48 curios for this number that have not yet been approved by an editor.)