122 Mr. C. J. Hargreave on the Law of Prime Numbers, 



Of the above quotients, the last sixteen only are greater than 

 the squares of their divisors. We have therefore to divide each ' 

 of these by the primes commencing with the divisor, and going 

 on to the prime next below the square root of the quotient. The 

 primes, counted as before from divisor to quotient, give the result 

 of this part of the process as being 8664. These divisions pro- 

 duce only three instances in which the quotient exceeds the 

 square of the divisor, and the repetition of the process upon these 

 gives 8. The result with reference to 1,000,000 is — 



No. of terms whose lowest prime factor exceeds 23 163,586 



Of which we find double numbers 76,428 



triple numbers 8,664 



quadruple numbers .... 8 



85,100 



78,486 



Adding the 9 excluded primes and deducting unity, we have the 



number of primes up to a million 78,494, being one more than 



Legendre counted. 



The application of the formula to 10,000,000 is much more 

 laborious. The results for 5 and 10 millions are as follows : — 

 For 5,000,000— 



No. of terms whose lowest factor exceeds 23 is 817,944 

 Of which we find double numbers .... 395,600 



triple numbers 72,900 



quadruple numbers . . . 925 



Leaving a final residue of primes .... 348,527^ 

 The number obtained by formula lia? is . . 348,634 

 The number given by Legendre's formula is 348,644 

 For 10,000,000 the results are- 

 No. of terms whose lowest factor exceeds 23 is 1,635,877 

 No. of double numbers included in these . 796,759 



No. of triple numbers included 170,827 



No. of quadruple numbers included . . . 3,667. 



Leaving a final residue of primes, (excluding unity, but including 

 the primes from 2 to 23), to the number of . . 664,632 

 The number obtained by the formula lia? is . . . 664,916 

 The number obtained by Legendre^s formula is . . 665,140. 

 These results siifficiently attest the truth of the formula which 

 I have previously demonstrated. Some years ago I obtained the 

 following results by the above process, but the calculations were 

 not made with so much care as those given in the present paper. 

 No. of primes up to 9,699,690 .... 645,544 

 No. of primes up to 4,849,845 .... 338,919 

 the corresponding results of the formula lia: being respectively 

 646,266 and 338,898. 

 Dublin, July 1,1864. 



