372 History of the Theory of Numbers. [Chap, xiv 
on dmding/^ by 2), 1+2, 1+2+2 under 1, 3, 5 of the first row for r = l; in 
the lower row, insert 4 (the quotient), 4 — 1, 4 — 2. To factor/^ — c, locate 
the column headed by the given c; thus, for c = 3, the factors are s = 6, 
r = 1 and s = 3, r = 2. Since c = 2 occurs only in the first row, 9 — 2 is prime. 
Joubin,^^^ J. P. Kulik,^^^ 0. V. Kielsen,i36 ^^d G. K. Winter^^e published 
papers not accessible to the author. 
E. Lucas gave methods of factoring and tests for primes (Ch. XVII). 
D. Biddle^" wrote the proposed number N in the form S^-\-A, where 
S- is the largest square < A^. Write three rows of numbers, the first begin- 
ning with A, or A —S if A >S; the second beginning with S (or S+1) and 
increasing by 1 ; the third beginning with S and decreasing by 1. Let A^, 
B„, C„ be the nth elements in the respective rows. Then 
except that, when i4„>C„, we subtract C„ from An as often (say k times) 
as will leave a positive remainder, and then B„ = 5„_i + 1+A:. WTien we 
reach a value of n for which i4„ = 0, we have N = BJJn- For example, if 
iV = 589 = 24H 13, the rows are 
13 14 17 1 9 
24 25 26 28 29 31 (factors 31, 19). 
24 23 22 21 20 19 
It may prove best to start with 2N instead of with N. 
O. Meissner^^^ reviewed many methods of factoring. 
R. W. D. Christie^^^ gave an obscure method by use of "roots." 
Christie"" noted that, if N = AB, 
A = {4hN+d''-d)/i2b), B = (46Ar+dHd)/2, d=a-hc, 
whence d'' = {B-h Ay. 
D. Biddle"^ gave a method of finding the factors of N given those of 
AT+l. SetL = iV-l. Try to choose i^ and M so that i(:iV/ = iV+ 1 and 
so that 1 +Z is a factor of N. Since 2N={l-\-K) M-\-L — M,we will have 
L-M=il+K)m, whence 2N={l+K){M+m). For iV=1829, A^+1 
= 2-3-5-61. Take K = 30, M = 61. Then m = 57, M+w = 2-59, iV = 31-59. 
He gave {ibid., p. 43) the theoretical test that N = S^+A is composite if 
the sum of r terms of 
is an integer for some value of r. 
'»*Sur lea facteura num6riques, Havre, 1831. 
>«Abh. K. Bohm. Gesell. Wiss., 1, 1841 (2, 1842-3, 47, graphical determination of primes). 
"HDra et heel tals upplosning i factorer, Kjobenhavn, 1841. 
"•Madras Jour. Lit. Sc, 1886-7, 13. 
"'Mesa. Math., 28, 1898-9, 116-20; Math. Quest. Educat. Timea, 70, 1899, 100, 122; 75, 1901, 
48; extension, (2), 29, 1916, 43-6. 
"«Math. Naturw. Blatter, 3, 1906, 97, 117, 137. 
"'Math. Quest. Educat. Times, (2), 12, 1907, 90-1, 107-8. 
"o/bid., (2), 13, 1908, 42-3, 62-3. 
"i/btd., (2), 14, 1908, 34. The process is well adapted to factoring 2P-1, (2), 23, 1913, 27-8. 
