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 and is the only number with both of these properties. 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 base 10, 3 is the only Mersenne prime ending in 3. All the other Mersenne primes end in either 1 or 7.
*Three is the base of the numeral system, also known as trinary.
== 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 \(...2,464,195,387\) (see moduli of power towers).
*In Greek- and Latin-based number naming systems, 3 is associated with prefix tri- (often shortened to tr-). For instance, the third -illion is called trillion.
*André Joyce used the Mayan word "ox", meaning 3, in some of his googologisms.
*Username5243 coined the name Ternary-Goonol for this number, and it's equal to 3[1]1 in Username5243's Array Notation.Part 1 - My Large Numbers
*In SAN, hyp_cos uses the number 3 to name googolisms.
=== 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: \(m_1(1) = 3\)
* gag: \(gag(1) = 3\)
* mag: \(mag(1) = 3\)
* Laver table: \(q(2)=3\)
* Gijswijt's sequence: \(c(2)=3\)
* Block subsequence theorem: \(n(1)=3\)
* Beklemishev's worms: \(\text{Worm}(1) = 3\)
* Mixed factorial: \(2^*=3\)
* H function: \(H(1)=3\)
* S function (both variants): \(S(1)=3\)
* U function: \(U(1)=3\)
== As a cash denomination ==
Some currencies, such as the and the , have banknotes with this number in the denomination.
Some currencies, such as the and the , have coins with this number in the denomination.
== Sources ==
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zh:三
Category:Numbers
Category:Banknote denominations
Category:Class 0
Category:Coin denominations
Category:Fibonacci numbers
Category:Numbers 0-99
Category:Palindromic prime numbers
Category:Prime numbers
 
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Category:Doogol Regiment
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