3

3 (three) is a positive integer following 2 and preceding 4. Its ordinal form is written "third" or "3rd".

Properties
3 is an, the smallest number with both of those properties. It is a   + 1 and a Mersenne prime   - 1. 3 is the 2nd.

In base 10, it is easy to test for divisibility by 3 by simply summing the digits of the number. The result will be the same modulo 3, so the process may be repeated. This is a consequence of 10 being one more than a multiple of 3.

In googology
BEAF arrays based on the number 3 are not degenerate, so Jonathan Bowers often coins googologisms based on this number. Some examples are tritri, tetratri, and dupertri. As the numbers get larger, their last digits converge to a single, very long string ending in \(...2464195387\) (see moduli of power towers).

In Greek- and Latin-based number naming systems, 3 is associated with prefix tri-.

Username5243 calls this number Ternary-Goonol, and it's equal to 2[1]1 in Username5243's Array Notation.

Googological functions returning 3

 * Xi function: \(\Xi(3)=3\)
 * Goodstein function: \(G(2)=3\)
 * Weak Goodstein function: \(g(2)=3\)
 * Kirby-Paris hydra: \(\text{Hydra}(2)=3\)
 * TREE function: \(\text{TREE}(2)=3\)
 * Fusible numbers: \(m_1(1) = 3\)
 * Laver table: \(q(2)=3\)
 * Gijswijt's sequence: \(c(2)=3\)
 * Block subsequence theorem: \(n(1)=3\)
 * Beklemishev's worms: \(\text{Worm}(1) = 3\)
 * H function: \(H(1)=3\)
 * S function (both variants): \(S(1)=3\)
 * U function: \(U(1)=3\)