121 (number)

Template:More citations needed Template:Infobox 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 Template:Mvar 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 ( $1 1^{2}$ ). But it cannot be expressed as the sum of any other number plus that number's digits, making 121 a self number.