1878.] Mr Glaisher, On factor tables. 133 



26,569 = 163 x 163 

 652= 4x163 



27,221 = 167 X 163 

 978= 6x163 



28,199 = 173 X 163 

 978= 6x163 



29,177 = 179 X 163 

 326 = 2 X 163 



29,503 = 181 X 163 



&c. &c. 



This process will give all the numbers to wliicli 163 belongs as 

 least factor up to (163/ = 4,330,747,- where the three-factor num- 

 bers commence. All that is required in order to reduce this to 

 mere addition is a list of differences of consecutive primes from 

 163 to j^l, I being the limit of the table, supposed less than 

 4,330,747, and a small table of even multiples of 163 from 

 2 X 163 to 2711 X 163, 2m being the greatest difference between 

 two consecutive primes between these limits. If I be 4,000,000, 

 the nearest prime below ygs ^ is 24,533 ; and the greatest differ- 

 ence is 52, between 19,609 and 19,661 \ The accuracy of the 

 work can be verified at any stage and as often as thought neces- 

 sary by multiplying together the two factors. Of course in the 

 calculation of the fourth million the commencement would be 

 made at 18,413 x 163 = 3,001,319, the smallest number exceeding 

 3,000,000 to which the least factor 163 belongs. 



We thus have two distinct methods, each of which has its special 

 advantages, viz. the sieve method and the method by calculation 

 of multiples. The latter is unsuitable for small primes, which 

 appear as least factors of numbers having three or more prime 

 factors; in fact this method is only appropriate for two-factor 

 numbers. On the other hand, the sieve method is rather more 

 suitable for the entry of small primes, as when the prime is large, 

 the great size of the sieve is inconvenient; this method points out 

 all multiples of the prime, not divisible by 2, 3, or 5, whether they 

 be two-factor, three-factor, four-factor, &c., numbers. 



It is clear that up to 163 the sieve method should be used ; 

 and that for 163 and beyond we may employ the multiple method. 

 Burckhardt states that he used the sieves for primes up to 500, 

 and the multiple method for higher primes. In the calculation of 

 the fourth million my father used sieves for -primes up to and 



1 The greatest difference between two consecutive primes np to 100,000 is 

 72 (31,397—31,469). For a list of the differences that exceed 50 and other allied 

 tables, see Messenger of Mathematics, yoI. yii. pp. 174—175 (March, 1878). 



10—2 



