i'll begin: 2
- The number two has many properties in mathematics. An integer is called even if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal and hexadecimal, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8. In numeral systems based on an odd number, divisibility by 2 can be tested by having a digital root that is even. Two is the smallest and first prime number, and the only even prime number (for this reason it is sometimes called "the oddest prime"). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and the first Smarandache-Wellin prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and a Markov number—appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers. It is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a superior highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six. Vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the base of the simplest numeral system in which natural numbers can be written concisely, being the length of the number a logarithm of the value of the number (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers. For any number x: x + x = 2 · x addition to multiplication x · x = x2 multiplication to exponentiation xx = x↑↑2 exponentiation to tetration In general: hyper(x,n,x) = hyper(x,(n + 1),2) Two also has the unique property that 2 + 2 = 2 · 2 = 22 = 2↑↑2 = 2↑↑↑2, and so on, no matter how high the level of the hyperoperation is.