I 
Chap. XIV] METHODS OF FACTORING. 373 
E. Lebon^^^" would first test AT for prime factors P just < VN. Let 
Q be the quotient and R the remainder on dividing N by P. If Q and R 
have a common factor, it divides N) if not, N is not divisible by any factor 
of Q or of R. 
D. Biddle^^^ considered N = S'^+A = {S+u){S-v), wrote w = iVi and 
obtained hke equations in letters with subscripts unity. Then treat 
UiVi = N2 similarly, etc. 
A. Cunningham^^^ noted that the number of steps in Biddle's^^^ process 
is approximately the value of A; in 2''' = N, and developed the process. 
E. Lebon^^^ treated the decomposition of forms 
a:"±a:^±a;T± . . .±1 (a>|8>7...) 
of degrees ^ 9 into two such forms, using a table of those forms of degrees 
^ 4 with all coefficients positive which are not factorable. The base most 
used in the examples is x = 10. But bases 2 and 3 are considered. 
E. Barbette^^^ quoted from his^^^ text the theorem that any integer N 
can be expressed in each of the four forms 
where ^j: = x{x-\-l)/2. The resulting new methods of factoring are now 
simplified by use of triangular and quadratic residues. The first formula 
implies N={x—y){x-\ry+\)/2. In his text, he considered the sum 
iv=(2/+i)+(y+2)+ . ..+{x-\)+x=^,-^y 
of consecutive integers. Treating four types of numbers N, he proved that 
this equation has 1, 2 or more than 2 sets of integral solutions x, y, according 
as iV is a power of 2, an odd prime, or a composite number not a power of 2. 
He proved independently, but again by use of sums of consecutive integers, 
that every composite number not a power of 2 can be given the form* N = u 
(2y — w+l)/2, where u and v are integers and v^u'^S. Solving for u, and 
setting x = 2v-\-l, we get 2u = x+(x'^ — 8Ny^^. Hence x'^—SN = y^ is solva- 
ble in integers [evidently by a: = 2A'+l, y = 2N—l]. Finally, Nz=Ax is 
equivalent to (2a;+l)^ = 8A^2;+l. For four types of numbers N, the solu- 
tions of y^ = SNz-\-l are found and seen to involve at least two arbitrary 
constants. 
A. Aubry^^^ reviewed various methods of factoring. 
"wil Pitagora, Palermo, 14, 1907-8, 96-7. 
i«Math. Quest. Educat. Times, (2), 19, 1911, 99-100; 22, 1912, 38-9; Educat. Times, 63, 1910, 
500; Math. Quest, and Solutions, 2, 1916, 36-42. 
"»/6id., (2), 20, 1911, 59-64; Educat. Times, 64, 1911, 135. 
i^BuU. soc. philomathique de Paris, (10), 2, 1910, 45-53; Sphinx-Oedipe, 1908-9, 81-3, 97-101 
"^L'enseignement math., 13, 1911, 261-277. 
"^Les sommes de p-i&mes puissances distinctes ^gales k une p-ieme puissance, Paris, 1910, 20-76. 
*This follows from the former result N = {x —y)(x+y +1) /2 by setting x=v, y = v—u. To 
give a direct proof, take u to be the least odd factor > 1 of the composite number N not 
a power of 2; then q = N /u can be given the form y — (u— 1)/2 by choice of y. If t;<M, 
then g<(«+l)/2<u, so that q has no odd factor and 2 = 2''. But N='2^U is of the 
desired form if we take v^u/2'^N. 
"'Sphinx-Oedipe, num^ro sp6c., June, 1911, 1-27. Errata and addenda, num6ro 3p6c., Jan., 
1912, 7-9, 14. L'enseignement math., 15, 1913, 202-231. 
