1879.] 



Separate Enumerations of Primes. 



195 



The results for the whole seven groups are 



Number of Number of Total 



4n + 1 4?z + 3 Difference. number of 



primes. primes. primes. 



24,758 24,616 142 49,374 



It should be stated that 1 was counted as a prime of the form 

 4?i + l ; 2, of course, was not counted at all. 



Tables showing the numbers of primes of the form 4?£ + l and of the 

 form 4?iH-3 in each ten thousand of the first hundred thousand 

 numbers of the first, second, third, fourth, seventh, eighth, and 

 ninth millions. 



First Million. 





Number of 

 4re + 1 

 primes. 



Number of 

 4» + 3 

 primes. 



Difference. 



Total number 

 of primes. 



0— 



10,000. . . . 



610 



619 



- 9 



1,229 



10,000— 



20,000. . . . 



516 



517 



- 1 



1,033 



20,000— 



30,000. . . . 



486 



497 



-11 



983 



30,000— 



40,000. . . , 



474 



484 



-10 



958 



40,000— 



50,000. . . . 



464 



466 



- 2 



930 



50,000— 



60,000. . . . 



469 



455 



+ 14 



924 



60,000— 



70,000. . . . 



431 



447 



-16 



878 



70,000— 



80,000. . . . 



454 



448 



+ 6 



902 



80,000— 



90,000. . . . 



445 



431 



+ 14 



876 



90,000— 



100,000. . . . 



435 



444 



- 9 



879 



0— 



100,000. . . . 



4,784 



4,808 



-24 



9,592 



Second Million. 





Number of 

 4n + l 

 primes. 



Number of 

 4rc + 3 

 primes. 



Difference. 



Total number 

 of primes. 



1,000,000—1,010,000. . . . 



390 



363 



+ 27 



753 



1.010,000—1,020,000 



367 



352 



+ 15 



719 



1,020,000—1,030,000. . . . 



369 



363 



+ 6 



732 



1,030,000—1,010,000 



347 



354 



- 7 



701 



1,040,000—1,050,000 



361 



370 



- 9 



731 



1,050,000—1,060,000 



349 



349 







698 



1,060,000—1,070,000 



362 



354 



+ 8 



716 



1,070,000—1,080,000. . . . 



372 



350 



+ 22 



722 



1,080,000—1,090,000. . . . 



352 



354 



— 2 



706 



1,090,000—1,100,000. . . . 



373 



365 



+ 8 



738 



1,000,000—1,100,000. . . . 



3,612 



3,574 



+ 68 



7,216 



