4

Properties

• 4 is a composite number, and the first natural number with this property. 4 is an even square number. 4 is the 2nd tetrahedral number. 4 is the 2nd Busy Beaver number. 4 is the 2nd superperfect number. Plugging 2 into the hyper operators degenerates to 4: \(2 \uparrow\uparrow ... \uparrow\uparrow 2 = 4\).
• 4 is associated with bad luck in Eastern numerology because the Chinese word for "four" sounds similar to the word for "death."
• It is possible to test for divisibility by 4 in base 10 by looking at last 2 digits of the number. If the 2-digit number is divisible by 4, then the whole number is.
• Ignoring weekdays, the Julian calendar repeats every 4 years.
• Its Roman numeral is most commonly IV, but on clock faces, IIII is also used.
• Its prime factorization is 4 = 2².
• All known multiperfect numbers > 6 are divisible by 4.
• Base four, sometimes used as a way of compressing binary, is called quaternary.

In googology

• Some googologisms based on 4 are supertet, quadroogol and googolquadriplex. 4 can be thought as zeroth term of sequence defining Graham's number: \(g_0\) = 4. 
• In Greek-based number naming systems, 4 is associated with prefix tetra-, and with prefix quadri- in Latin systems.
• 4 can be named gartwo, fztwo, fugatwo, megafugatwo, or boogatwo with the gar-, fz-, fuga-, megafuga-, and booga- prefixes respectively.
• Aarex Tiaokhiao coined the name binary-booiol for this number.^[1]
• Username5243 coined the names Binary-Goodol and Binary-Goonolplex for this number, and it's equal to 2[1]2 = 2[1]2[1]1 in Username5243's Array Notation.^[2]

Googological functions returning 4

• Rado's Sigma Function: \(\Sigma(2)=4\)
• Xi function: \(\Xi(4)=4\)
• Hyperfactorial: \(H(2)=4\)
• Factorexation: \(2\,\backslash=4\)
• Superfactorial (Clifford Pickover): \(2\$=4\)

Sources

1. ↑ Part 1 (LAN) - Aarex Googology
2. ↑ Part 1 - My Large Numbers