120 Mr. C. J. Hargrcave on the Law of Prime Numbers. 



eliminating those numbers whicli contain the iirst 10 or 11, 

 or at the utmost 12 primes. I therefore pass to another pro- 

 cess. 



If we take any number, say 10,000, and divide it successively 

 by the prime numbers less than its square root, and if we count 

 the number of primes between each divisor and its quotient 

 (both included when prime), the aggregate of these results is 

 obviously the number of numbers up to 10,000 which are com- 

 posite of two primes, or double numbers. For each prime from 



nn 



Pi to — has a corresponding double number, of which p^ is one 



Pi 

 factor and the prime in question is the other ; so that this pro- 

 cess exactly exhausts all the double numbers including squares. 

 If now we take each quotient obtained in the last process, down 

 to that quotient which was obtained by dividing by the prime 

 next below the cube root of w, and deal with it in exactly the 

 same manner as we before dealt with the number itself, that is, 

 divide it by every prime up to its square root, and count the 

 primes from the divisor to the quotient, the aggregate of the 

 results will exactly exhaust all the composites of three primes or 

 treble numbers including cubes. The repetition of the operation 

 upon each of the quotients in the last part of the process, or 

 rather upon such of them as admit of the operation, gives us all 

 the quadruple numbers ; and so on, as far as the magnitude of 

 the number enables us to carry the process. 



The number of prime divisors diminishes as we proceed, but 

 the number of dividends to be operated upon increases rapidly ; 

 so much so, that this process is utterly impracticable for those 

 small prime divisors which occur early in the series, such as 2, 

 3, 5, &c. 



We have, therefore, two processes ; the first of which enables 

 us to expel that large mass of composite numbers which con- 

 tain the small primes, such as from 2 to 19 or 23, or if need 

 be, to 29 or even 31, but is scarcely practicable beyond this 

 point ; and the second of which processes enables us to ascertain 

 the number of composite numbers which include only the larger 

 primes, such as those lying between 19 or 23 and the square 

 root of the number, but which would be quite unavailing for the 

 determination of the number of terms involving the small primes 

 2, 3, 5, &c. By using both processes, however, we bring the 

 problem within the province of reasonable industry. 



As the second process may not be veiy clear without an ex- 

 ample, I will apply the two processes to the determination of the 

 number of primtb under one million, for which we shall not have 

 occasion to use any table of primes beyond 35,000. 



Using the first process for the primes from 2 to 23, we find 



