180 (number) two square numbers: 12 + 6. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3. 180 is a 61-gonal number. Half a circle has 180 degrees. Summing Euler's totient function φ("x") over the first + 24 integers gives
Prime number theorem on primes in arithmetic progressions as a special case. In the theory of finite groups the Sylow theorems imply that, if a power of a prime number formula_195 divides the order of a group, then it has a subgroup of order formula_195. By Lagrange's theorem, any group of prime order is a cyclic group, and by the Burnside theorem any group whose order is divisible by only two primes is solvable. For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications