CHAP. XIV] METHODS OF FACTOBING. 373 E. Lebon1410 would first test N 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. Biddle142 considered N=*S2+A = (S+u)(S-v), wrote uv**^ and obtained like equations in letters with subscripts unity. Then treat UiVi~N2 similarly, etc. A. Cunningham143 noted that the number of steps in Biddle's142 process is approximately the value of k in 2*=JV, and developed the process. E. Lebon144 treated the decomposition of forms 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. Barbette145 quoted from his146 text the theorem that any integer N can be expressed in each of the four forms where Ax=x(rc+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+y+!)/2. In his text, he considered the sum 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 TV 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 giv.en the form* N = u (2v— ii+l)/2, where u and v are integers and v^.u*z.3. Solving for u, and setting x = 2v-\-l, we get 2u=x-\-(x2— &N)l/z. Hence x*—8N = y~ is solvable in integers [evidently by x = 2N+l, y = 2N— 1]. Finally, Nz=&T is equivalent to (2z+l)2 = 8Nz+l. For four types of numbers N, the solutions of y2 = 8Nz+l are found and seen to involve at least two arbitrary constants. A. Aubry147 reviewed various methods of factoring. 14iall Pitagora, Palermo, 14, 1907-8, 96-7. 142Math. 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. ™IUd., (2), 20, 1911, 59-64; Educat. Times, 64, 1911, 135. l44Bull. soc. philomathique de Paris, (10), 2, 1910, 45-53; Sphinx-Oedipe, 1908-9, $1-3, 97-101 146L'enseignement math., 13, 1911, 261-277. 146Les eommes de p-iemes puissances distinctes e'gales a une p-ieme puissance, Paris, 1910, 20-76. This follows from the former result N**(x— y)(a;-fy+l)/2 by setting x*=r, 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 w— (u— 1)/2 by choice of y. If r<n, then q<(u+l)/2<u, so that q haa no odd factor and g-2\ But N-2hU is of the desired form if we take v «»u/2 — JV. 147Sphinx-Oedipe, nume'ro spe*c., June, 1911, 1-27. Errata and addenda, nume>o sp^c., Jan., 1912, 7-9, 14. L'enseignement math., 15, 1913, 202-231.