121 (number)

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

In mathematics

121 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 {\displaystyle 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 {\displaystyle 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 {\displaystyle 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 {\displaystyle 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 {\displaystyle 11^{2}} ). But it cannot be expressed as the sum of any other number plus that number's digits, making 121 a self number.

References