1879.] 



Separate Enumerations of Primes. 



193 



numbers, viz., the subjoined tables give the number of primes of the 

 form 4?i-j-l and the number of primes of the form 4w + 3 in each ten 

 thousand of the hundred thousand numbers of the first, second, third, 

 fourth, seventh, eighth, and ninth millions. The properties of the 

 4n + l primes and the 4^ + 3 primes are very different ; and the idea 

 of separately enumerating the two forms originated with Professor 

 Tchebycheff, and was conveyed to me through M. Edouard Lucas. 



The enumerations were effected as follows : — Each page of Burck- 

 hardt's and Dase's tables contains eighty lines corresponding to the 

 eighty numbers 1, 7, . . . 97, 101, 103, . . . 199, 203, 209, . . . 299, i.e., 

 to the eighty numbers less than 300, and not divisible by 2, 3, or 5. 

 The lines corresponding to numbers of the form 4r^ + l were ruled 

 through in pencil, and then the bars (representing primes) which 

 were ruled over were enumerated separately, and also the bars which 

 were intact. The enumerations for the first three millions were made 

 from Burckhardt's tables, and those for the seventh, eighth, and ninth 

 millions from Dase's tables. In the case of the fourth million the 

 enumerations were made from the proof sheets of my father's factor 

 table of this million, the whole of which is now printed, and, with the 

 exception of the few last pages, stereotyped. 



The results for the groups in the first three millions were commu- 

 nicated to the Meeting of the British Association at Dublin, and are 

 published on p. 471 of the Beport for 1878. It is there stated that 

 " the numbers given in the table are the result of a duplicate enume- 

 ration ; but a third enumeration will be required, in order to render it 

 certain that they are absolutely free from error." This third enume- 

 ration has now been made and the following two errors in the British 

 Association table were detected by means of it : in the first ten thou- 

 sand of the second million the numbers of 4?i + l and 4^ + 3 primes 

 should be respectively 390 and 363, instead of 391 and 362 as printed, 

 and in the third ten thousand of the third million the numbers should 

 be 350 and 343 instead of 349 and 344 as printed. In the case of the 

 other four millions, the results for which are here given for the first 

 time, I have thought a careful duplicate calculation sufficient. 



The number of 4% + l primes in each hundred was found by count- 

 ing, and entered in its place on a printed form; and the number of 

 4^ + 3 primes was similarly found and entered on another form. 

 As a verification, the two results for each hundred were added 

 together, and entered on a third form. These totals should agree 

 with the numbers of primes in each hundred, which had been obtained 

 in the previous enumeration that I had made of all the primes in the 

 seven millions ; and such was found to be the case. It is to be 

 observed, however, that any error arising from the inaccurate or 

 imperfect ruling of a pencil line would not be detected in this 

 manner, as such an error would consist in a 4?i + l prime being 



