400 (number) + 61 + 67), strictly non-palindromic number 440 = 2 × 5 × 11, the sum of the first seventeen prime numbers, Harshad number, 441 = 3 × 7 = 21 442 = 2 × 13 × 17, sphenic number, sum of eight consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71) A prime number, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part, Mertens function sets new low of -9, which stands until 659. 444 = 2 × 3 × 37, refactorable number, Harshad number. 445 = 5 ×
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