ON MATHEMATICAL TABLES. 175 



multiple of 13 exceeding 3,000,000 is 3,000,023 : add 26 continually till 

 3,000,000 -f- 13 x 300 is reached, and then throw out the multiples of 3 

 and 5 ; there are thus left 80 numbers, which correspond to the squares 

 to be cut out from the sieve. The accuracy of the 80 numbers that 

 remain was verified by differencing them ; as the differences recur with a 

 period of eight.* 



In general the sieve for the prime p contains p 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, 3, or 5, of the form 300 q + a, where a is any one of the 

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

 p be prime to r, and if p, 2p, 3p,. . . (r— 1) p be divided by r, the remainders 

 are the r — 1, numbers 1, 2, 3,...r— 1 ; in this case, therefore, if p, 2p, 3p, 

 ...299p be divided by 300, the remainders are the 299 numbers 1, 2, 3,... 

 299, and if 2p, 3p,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 3002 + a be divisible by^>, the next number in 

 the same line divisible by p is 300qp + a, viz.,. is a number^ columns fur- 

 ther on. 



The cube root of 4,000,000 is 158 - 74..., and in a factor table extend- 

 ing to 4,000,000, 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 commence- 

 ment, the least factor 163 first appears at the number 163 x 163, then at 

 167x163, 173x163, 179x163, 181x163, &c. ; 163, 167, 173, 179, 181, 

 &c, being the series of primes starting from 163 ; for we only consider 

 products of two primes, of which 163 is the smaller, that is, numbers 

 formed by multiplying l63 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 — 



26,569 = 163 x 163 

 652 = 4 x 163 



27,221 = 167 x 163 

 978 = 6 x 163 



28,199 = 173 x 163 

 978 = 6 x 163 



29,177 = 179 x 163 

 326 = 2 x 163 



29,503 = 181 x 163 

 &c. «fec. 



* It is easily seen that this must be so ; for form the multiples of the prime p 

 that are not divisible by 2, 3, or 5; these are p, Ip, Up, 13/>, Up, 19p, 23p, 29p, 

 then the next eight are obtained by adding 30/; to each of these and so on. Thus 

 the differences are 6p, ip, 2p, ip, 2p, 4p, 6p, 2p, recurring with a period of eight. 



