121 (number)

121 (one hundred [and] twenty-one) is the natural number following 120 and preceding 122.

In mathematics

One hundred [and] twenty-one is

a square (11 times 11)
the sum of the powers of 3 from 0 to 4, so a repunit in ternary. Furthermore, 121 is the only square of the form $1+p+p^{2}+p^{3}+p^{4}$ , where p is prime (3, in this case).^[1]
the sum of three consecutive prime numbers (37 + 41 + 43).
As $5!+1=121$ , it provides a solution to Brocard's problem. There are only two other squares known to be of the form $n!+1$ . Another example of 121 being one of the few numbers supporting a conjecture is that Fermat conjectured that 4 and 121 are the only perfect squares of the form $x^{3}-4$ (with $x$ being 2 and 5, respectively).^[2]
It is also a star number, a centered tetrahedral number, and a centered octagonal number.

In decimal, it is a Smith number since its digits add up to the same value as its factorization (which uses the same digits) and as a consequence of that it is a Friedman number ( $11^{2}$ ). But it cannot be expressed as the sum of any other number plus that number's digits, making 121 a self number.

121 is also:

The electricity emergency telephone number in Egypt
The number for voicemail for mobile phones on the Vodafone network^[3]
The undiscovered chemical element unbiunium has the atomic number 121
The official end score for cribbage^[4]
The pennant number of RTS Moskva, the Russian Navy’s Black Sea flagship, which was damaged beyond repair on April 13, 2022.

^ Ribenboim, Paulo (1994). Catalan's conjecture : are 8 and 9 the only consecutive powers?. Boston: Academic Press. ISBN 0-12-587170-8. OCLC 29671943.
^ Wells, D., The Penguin Dictionary of Curious and Interesting Numbers, London: Penguin Group. (1987): 136
^ Vodafone, Calling and messaging
^ Rule 1.1 Archived 2015-01-18 at the Wayback Machine, American Cribbage Congress, retrieved 6 September 2011