132 Mr Glaisher, On factor tables. [Feb. 11, 



empty square write tlie number 13. Then place the sieve over 

 the next 13 columns and proceed as before, and so on throughout 

 the whole 44 sheets. 



The sieve for the next prime, 17, contains 17 columns, and is 

 made in the same way, viz. by cutting out the squares correspond- 

 ing to the numbers between 3,000,000 and 3,000,000 + 17 x 300, 

 which are divisible by 17, and not by 2, 3, or 5. Then this 

 sieve is placed over the first 17 columns, and 17 entered in 

 all the empty squares, then placed over the next 17, &c., and 

 so on. 



In general the sieve for the prime jo contains jj columns, and 

 it is to be noted that every sieve, whatever its length, has 

 exactly 80 squares cut out, one in each line. To show that there 

 must be one square cut 'out in each line it is only necessary 

 to observe that p must have some multiple, not divisible by 2, 

 8, or 5, of the form 300^' + a, where a is any one of the 80 

 numbers less than 300 and prime to it. For, by a known 

 theorem, if ^ be prime to r, and if jj, 2j), Sp,...{r — l)p be divided 

 by r, the remainders are the r^l numbers 1, 2, 3, ... 7' — 1 ; in 

 this case, therefore, if^, 2p, Sp ... 299p be divided by 300, the re- 

 mainders are the 299 numbers 1, 2, 3,... 299, and if 2p, Sp, 4p,... 

 and all the multiples of p divisible by 2, 3, or 5 be thrown out, 

 the remainders divisible by 2, 3, or 5 are thrown out also, and the 

 remainders left are the 80 numbers less than 300 and prime to it. 

 Also, there cannot be two squares in the same line cut out from 

 the sieve, for a being a given number, if oOOq + a be divisible 

 by p, the next number in the same line divisible by p is SOOqp + a, 

 viz. is a number p) columns further on. 



The cube root of 4,000,000 is 15874..., so that the prime 157 

 appears once, and only once, as the least factor of a three- 

 factor number, viz. for 3,869,893. Thus 163 and larger primes 

 will only occur as least factors of two-factor numbers, and we 

 may find the numbers to which they belong without the use of the 

 sieves as follows : 



Supposing that we are constructing a factor table from the 

 commencement, the least factor 163 first appears at the number 

 163 X 163, then at 167 x 163, 173 x 163, 179 x 163, 181 x 163, 

 &c.; 163, 167, 173, 179, 181, &c. being the series of primes start- 

 ing from 163 ; for we only consider products of two primes, of 

 which 163 is the smaller, that is, numbers formed by multiplying 

 163 by the primes greater than itself. To obtain the results of 

 the multiplications it is only necessary to add to 163 x 163 the 

 product 4 X 163, and to this 6 x 163, &c. ; the work standing 

 thus 



