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 a quadratic polynomial that (for integer arguments) is prime infinitely often. Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes, Euler showed that, for any arbitrary real number formula_22, there exists a prime formula_20 for which this sum is bigger than formula_22. This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every formula_22. The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum does not