268 



REPORT — 1879. 



9. Note on the Enumerations of Privies of the Forms 4m + 1 and 4m + 3. 

 By J. W. L. Glaisher, M.A., F.B.8. 



At the last meeting of the British Association I communicated the results of an 

 enumeration, then just completed, of the primes of the form 4m + 1 and of the 

 form 4m + 3 in three groups, each of 100,000 numhers, viz., between and 100,000, 

 between 1,000,000 and 1,100,000, and between 2,000,000 and 2,100,000. These 

 results are printed on page 471 of the ' Keport ' for 1878. It is there stated that 

 ' the numbers given in the table are the result of a duplicate enumeration ; but a 

 third enumeration will be required, in order to render it certain that they are 

 absolutely free from error.' This third enumeration has now been made, and the 

 following two errors in the table were detected by means of it : in the first ten 

 thousand of the second million the numbers of 4m + 1 and 4m + 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. The totals of the columns thus become 3,642 and 3,574 in 

 the second million, and 3,463 and 3,411 in the third million. 



Since the meeting at Dublin the enumerations have also been made for the first 

 hundred thousand numbers of the fourth, seventh, eighth, and ninth millions. In 

 the case of the fourth million the enumerations were made from the proof sheets 

 of my father's factor table for this million, which is now stereotyped and ready for 

 publication. 



The total numbers for primes of the forms 4m -i- 1 and 4n + 3 in the first one 

 hundred thousand numbers of each of the seven millions are — 



The results for the whole seven groups are — 



Number of 



4re + 1 primes 



24,758 



Number of 



4n + 3 primes 



24,616 



Difference 

 142 



Total number 



of primes 



49,374 



It should be stated that 1 was counted as a prime of the form 4m + 1 ; 2, of 

 course, was not counted at all. 



The details of these enumerations will appear in the 'Proceedings of the 

 Royal Society ' (vol. xxix. pp. 192-197). 



Professor Tchebycheff, in a letter to M. Fuss, ' sur un nouveau th<5oreme relatif 

 aux nombres premiers contenus dans les formes 4m + 1 et 4m + 3,' * states that he 

 has found that the functions which determine the total number of primes of the 

 form 4m + 1 and the total number of those of the form 4m + 3, inferior to a given 

 very large limit x, differ essentially in their second terms ; this term being 

 greater in the latter case than in the former, so that for certain values of x the 

 number of 4m + 3 primes exceeds that of 4m + 1 primes by a number approximately 



equal to -, • 



* log x 



Of course an enumeration of primes in certain groups such as those chosen 



1 ' Bulletin de la Classe Physico-Mathematique de l'Academie Imperiale des 

 Sciences de Saint-Petersbourg,' t. xi. (1853), col. 208. 



