Chap. XIV] 
Methods of Factoring. 
371 
F. Landry ^^^ 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 n+n' = 6/i+r. 
Qnn'+n-\-n' = — - — = Qq-\-r. 
6 
Then nn' = q — h, whence 
- q—n'(r—n') 
h = - 
6n' + l 
Give to n' values such that 6n'+l is a prime < ViV. 
K. P. Nordlund^25 treated Qp-l = {Qm+l){Qn-l) solved for m. 
D. Biddle^^^ applied the method to 6n±l. 
Hansen,^^ of Ch. XIII, used this method. 
Miscellaneous Methods of Factoring. 
Matsunaga^^^ wrote the number to be factored in the form r^+R. For 
r odd, set r = Bi, Bi — 2 = B2, Bg — 2 = ^3, ... and perform the following 
calculations : 
R = Q,B,+Au 
Ai+K2' = Q2B2+A2, 
A2-\-Ks' = Q3Bs+As, 
etc., until we reach A„ = 0; then Bn is a factor, 
and replace i? by jR + 1 in what precedes. 
J. H. Lambert ^^° used periodic decimals [see Lambert,^ Ch. VI]. 
Jean Bernoulli^^^ gave a method based on that of Lambert (Mem. de 
Math. Allemands, vol. 2). Let A=a^+b have the factors a—x and 
a+x+y. Then x^ = ay— xy — b. Solve for a:. Thus 2/^+ 4a?/ — 46 must be 
a square. Take y = l, 2, . . . and use a table of squares. 
J. Gough^^^ gave a method to find the factors r, s of each number f^ — c 
between (/— 1)^ and/^. 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, (/— 1)^, . . ., /^. In the upper row for r = 2, insert 1 (the remainder 
K2 = 2Q2-\-K2', 
K3 = 2Qs+Ks', 
K2'^^K,+4, 
K3' = 7^2+8, 
If r is even, set r — 1 = Bi 
r = l 
c = 5 
s = 4 
4 
5 
3 
6 
2 
7 
1 
8 

9 
r = 2 
c = 5 
s = 2 
3 
3 
1 
4 
r = 3 
c = 
s = 

3 
i2'»Assoc. fran?. avanc. sc, 9, 1880, 185-9. 
i25Nyt Tidsskrift for Mat., Kjobenhavn, 15 A, 1904, 36-40. 
i^^Math. Quest. Educ. Times, 69, 1898, 87-8; (2), 22, 1912, 38-9, 84-6. 
i^'Japanese manuscript, first half eighteenth century, Abhandl. Geschichte Math. Wiss., 30, 
1912, 236-7. isoNoya Acta Eruditorum, 1769, 107-128. 
"iNouv. Mem. Ac. BerHn, ann6e 1771, 1773, 323. 
"2Jour. Nat. Phil. Chem. Arts (ed., Nicholson), 1, 1809, 1-4. 
