Taking the third power of a number is called taking its cube.
Three is the only number to be both a Mersenne prime and a Fermat prime. It is also a Mersenne prime exponent producing the prime 7. The next exponent is 5, the next Fermat prime is 5, and the next Mersenne prime is 7
3 is in a twin prime pair with 5.
Three is the second lucky number, between 1 and 7.
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.
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).
André Joyce used the Mayan word "ox", meaning 3, in some of his googologisms.
Testing if a number is a multiple of 3 is as easy as adding up the digits and seeing if they add to a multiple of 3. So 59604 is a multiple of 3 since its digits add to 24, which is a multiple of 3.