414 



NATURE 



\Feb. 28, i: 



applied to find the factors of 3979769, instead of "hundreds of 

 division sums " being requisite, — about two minutes of time, 

 no division sum at all, only one of subtraction, showed them to 

 be 201 I X 1979. 



My method or rule consists in finding the next higher square 

 (call it " A ") to the proposed number, from which, if the 

 proposed number be deducted, the difference, or remainder, will 

 be a square number (call it " B "). Then the square root of 

 A plus the square root of B will be one factor, and the square 

 root of A minus the square root of B will be the other. 



It is essential to state, in addition to this general rule, that, 

 when the "difference" between the proposed number and the 

 higher square first used (not necessarily the next higher square 

 to it) is a square, the process is virtually ended, for the factors 

 can then be readily found by the directions given above. But if 

 "the difference" be not a square, successive additions must be 

 made to it of a progressive series of numbers, whose common 

 difference is 2, and commencing with twice the square root of 

 the square first used, plus 1, until their sum becomes a square 

 number, which will be that called B in the paragraph above. 



This is simply an easier mode of ascertaining what the differ- 

 ences may be between the proposed number and the higher 

 squares, than by subtracting one from the other at each step of 

 the process. The aggregate sum of the additions, taken at any 

 step, is always equal to the square of the square root correspond- 

 ing to it, minus the proposed number. The square root cor- 

 responding to any step is always half the serial number then 

 added, plus \. 



Hence my method may also be said to consist in the successive 

 addition to the difference between any proposed number and a 

 higher square, of a series of numbers as specified above, until 

 their sum becomes a square. 



The length of the process varies, and is longest when the 

 difference between the factors of the proposed number is 

 greatest, and especially if it be a high prime. But in many 

 cases it can be shortened very considerably. It would require 

 many examples to show how this can best be done under many 

 varying circumstances. At present I give only a few examples 

 to show the operation according to the general rule, and of one 

 or two ways of shortening the work. 



Examples. - 

 and 3979769- 



-Find factors of 1443, 57, 1 10467, 8616460799, 



Proposed number 

 Next higher square 38^ 



1443 

 1444 



Difference 



Then 



38 38 



Factors are ... 39 x 37 = 1443- 



Proposed number 57 



Next higher square (8-) 64 



Difference ... 7 



Add 8 X 2 -t- I (or 8 -t- 9) ... 17 = 24 



,, 9 X 2 -t- I (or 9+10) ... 19 = 43 



,, 10 X 2 + I (or 10 -H II) ... 21 = 64 = 8- 



Then 11 n 



-t-8 -8 



Factors are ... 19 x 3 = 57. 



Proposed number 11 0467 



Next higher square (333-) 1 10889 



Difference ... 422 



Add 333 X 2+ I (or 333 -t- 334)... 667 



1089 = 33^ 



334 3.-?4 

 ^li -33 



Factors are... 367 x 301 = 110467. 



Jevons's proposed number ... 

 Next higher square (92825^) ... 



8616460799 

 8616480625 



Difference ... 

 Add 92825 X 2 -f I (or 92825 + 92826) 



,, 92826 X 2 -t- I 



19826 

 18565 1 = 205477 

 (not a sq.) 



185653 =391130 

 (not a sq.) 



therefore, add fifty-four more serial numbers, to 92880 x 2 -f i ; 



the sum of the additions will then be found to correspond with — 



92880^ = 8626694400 1 _ 



- 8616460799 / 



Then 92880 92880 



+ 3199 -3199 



1 023360 1 = 3199^ 



Factors are 



96079 X 89681 = 8616460799. 



" C.W.M.'s " proposed number 

 Next higher square (1995-) ... 



3979769 

 3980025 



Then 



Factors are 



Difference 



1995 1995 

 16-16 



256 = 16^ 



1979 = 3979769- 



To find the lowest factois of 12267- 

 Proposed number ... 



12267 



Sum of digits being 18 it is divisible by 3'^ = 9 

 Next higher square (37-) , 



Difference 



Add 37 X 2 -1- I or 37 + 38 



Then 



Lowest factors. 



38 

 + 9 



47 



1363 



1369 



6 

 75 



81 ■■ 



3 = 12267. 



To find the factors of 73, by the general rule, 28 steps in the 

 process are requisite, until the sum of additions to first difference 

 reaches a square, 1296 — 36- = 37" - 73. 



Then 37 -h 36 



,_ _ g _ '^ I the only factors, therefore 73 



a prime. 



Instead of 28 steps being taken, the process may_be shortened 

 thus : — 



It must be noted that the first difference (8) is an even 

 number, and that the second (27) is an odd one. In line with 

 27, put down difference (22) between it and next higher odd 

 square (49), and in line with 8, the difference (28) between it and 

 the even square (36) next below 49. As the difference 6 

 (between 28 and 22) will not divide either without a remainder, 

 the process must be repeated, with the succeeding higher even 

 (64) and odd (81) squares, for each line, until we obtain two 

 numbers, divisible by their difference, without remainder. In 

 this example we find that the second step gives what is 

 required. The difference (2) between 56 and 54, divides 

 either. 



Then 



and 372 = 13691 

 - 73/ 



28, and 28-1- 9 (9 is first square root used) = 37 ; 

 27, and 27 -t- 10 (10 is second sq. root used) = 37 ; 

 16 = 36-, as stated above. 



