J 76 report— 1878. 



This process will give all the numbers to which 163 belongs as least 

 factor up to (163) 3 = 4,330,747, where the three-factor numbers commence. 

 All that is required in order to reduce this to mere addition is a list of 

 differences of consecutive primes from 163 to y^Z, 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 2m X 163, 2m being the greatest difference between 

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

 prime below j^l is 24,533 ; and the greatest difference is 52, between 

 19,609 and 19,661.* The accuracy of the work can be verified at any 

 stage and as often as thought necessary by multiplying together the two 

 factors. Of course in the calculation of the fourth million the commence- 

 ment 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. 



There are thus two distinct methods, each of which has its special 

 advantages, viz., the sieve method and the method by calculation of multi- 

 ples. 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, however, 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 sieves for primes up to 500, and the multiple method for 

 higher primes. In the calculation of the fourth million sieves were used 

 for primes up to and including 307, and the multiple method was employed 

 for primes from 211 to 1999. The numbers corresponding to the least 

 factors from 211 to 307 inclusive were obtained by both methods. 



As the multiple method only gives numbers where the least factor is 

 the given prime p, it follows that every number so found must correspond' 

 to an empty square, and the verification thus afforded of the entries 

 already made was very valuable. 



The sieve for 307 contains 307 columns, and therefore occupies four 

 sheets all but one column : considered as a whole, therefore, it has only 

 to be moved 11 times for the million, while the sieve for 13 has to be- 

 moved 257 times.f 



Before the calculation was begun, it seemed as if the excessive length 



* The greatest difference between two consecutive primes up 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,' vol. vii. pp. 174-175 (March, 1878). 



f In the fourth million the 13's were entered by a sieve consisting of 13 columns, 

 the 17's by a sieve of 17 columns, and so on. In the fifth and sixth millions now in 

 progress, the 13*s were entered by a sieve of 78 columns, equivalent to six 13-sieves 

 fixed together. This was found to greatly facilitate the entries, as the number of 

 removals of the sieve was reduced in the proportion of 6 to 1, and there was less 

 risk of error. The saving of time effected by the use of the 78-column sieve 

 amounted to nearly one-half. For the 17's a sieve of 5 x 17, = 85, columns was 

 used, for the 19's a sieve of 4 x 19, = 76, columns, and so on, the number of columns 

 being made as nearly as possible equal to the number of columns (77) on a sheet. 

 It was found also that by the use of the long sieves the sheets were much better 

 preserved from wear and tear, as the sheet upon which the factors were being 

 entered was in general almost wholly covered by the sieve, and so protected from 

 friction, &c. 



