CHAP. XIV] METHODS OP FACTORING. 371 F. Landry124 treated the possible pairs 6n=*= 1 and 6n'=±= 1 of factors of N. Taking for example the case of the upper signs, we have Set o Then nn' = q-h, whence 6n'+l Give to n' values such that 6n'-f 1 is a prime <' K. P. Nordhmd126 treated 6p -1 = (6w+l) (6n-1) solved for m. D. Biddle126 applied the method to 6n=*=l. Hansen," of Ch. XIII, used this method. MISCELLANEOUS METHODS OF FACTOKING. Matsunaga129 wrote the number to be factored in the form r*+R. For r odd, set r=Bly B1—2 = B2) B2 — 2=B3)... and perform the following calculations: etc., until we reach An = 0; then Bn is a factor. If r is even, set r —1 =B^ and replace .K by R+l in what precedes. J. H. Lambert130 used periodic decimals [see Lambert,6 Ch. VI]. Jean Bernoulli131 gave a method based on that of Lambert (Me*m. de Math. Allemands, vol. 2). Let ^L=a2-f6 have the factors a—x and a+x+y. Then x2 = ay—xy — b. Solve for x. Thus i/2+4ai/ —46 must be a square. Take # = 1,2,... and use a table of squares. J. Gough132 gave a method to find the factors r, s of each number jf2—c between (/—I)3 and/2. For example, let-/=3 and make a double row for each r = l,...,/. In the upper row for r = l, insert 2/—1,..., 1, 0; in the lower, (/— I)2,..., /2. In the upper row for r=2, insert 1 (the remainder c=5 4 3 2 1 0 r 1 s^4 5 6 7 8 9 c = 5 3 1 r-2 s = 2 3 4 c — 0 s = 3 mAssoc. frang. avanc. sc., 9, 1880, 185-9. 126Nyt Tidsskrift for Mat., Kjobenhavn, 15 A, 1904, 36-40. ^Math. Quest. Educ. Times, 69, 1898, 87-8; (2), 22, 1912, 38-9, 84-6. 129Japanese manuscript, first half eighteenth century, Abhandl. Geschichte Math. Wiss., 30, 1912, 236-7. "°Nova Acta Eruditorum, 1769, 107-128. 131Nouv. Mem. Ac. Berlin, annee 1771, 1773, 323. "2J0ur. Nat. Phil. Chem. Arts (ed., Nicholson), 1, 1809, 1-4.