^'11 Chap. XIV] METHODS OF FACTORING. 367 
A0 = 10x^ — 12x — 4:, whence 12a:+4 is divisible by 10, so that a: = 5(i — 2. 
Then A=25d'^ — 2Qd+Q. Thus if we delete the last two digits 7 , 6 of squares 
A7Q, we obtain numbers A whose values for d = l, 2, . . . can be derived 
from the initial one 5 by successive additions of 49, 49+50, 49+2-50, . . . . 
He gave such results for every pair of possible endings of squares. 
A similar method is applied to any composite number. One case is 
when the last two digits are m, 1 and Aml = (10a; — 1)(10?/ — 1). Then 
A0 = 10xy — y — x — m, y+x+m = 10a, A = 10ax—x'^—mx — a. 
The discriminant of the last equation must be a square. A table of values 
of A for each a may be formed by successive additions. 
G. Speckmann^^ noted that the two factors of AT" = 2047 end in 1 and 7 
or 3 and 9. Treating the first case, we see that, if a and b are the digits in 
tens place, 6+7a=4 (mod 10), so that the factors end in 01 and 47, or 11 and 
77, etc. 
G. Speckmann^^ wrote the given number prime to 3 in the form 9a +6 
(6<9), so that the sum of its digits is =6 (mod 9). By use of a small 
auxiliary table we have the residues modulo 9 of the sums of the digits of 
every possible pair of factors. 
R. W. D. Christie^^ and D. Biddle^° made an extensive use of terminal 
digits. 
E. Barbette^ ^ noted that lOd+u has a divisor lOw — 1 if and only if 
d+?nw has that divisor. Set d-\-'mu = n(10m — l), d = 10d'+u'. Then 
mn = d'-\-x, lOx = 'mu-\-n-\-u'. 
Eliminating n, we get a quadratic for m. Its discriminant is a quadratic 
function of x which is to be made a square. Similarly for lOm + 1, 10m ±3. 
A. Gerardin^^" developed Barbette's^^ method. 
R. Rawson^^ found Fermat's^ factors of a number proposed by Mersenne 
by writing it to the base 100 and expressing it as (a- 10^ +23) (6 -10^ +3). 
J. Deschamps^^ would use the final digits and auxiliary tables. 
A. Gerardin®^ would factor N (prime to 2, 3, 5) by use of 
iV=120n+i^=(120x+a)(120i/+6), 
and a table showing, for each of the 32 values of K< 120, the 16 pairs o, b 
(each< 120) such that ab=K (mod 120). He factored Mersenne's number.^ 
Factoring by Continued Fractions or Fell Equations. 
Franz von Schaffgotsch^"" would factor a by solving az^-^l = x^ (having 
"Archiv Math. Phys., (2), 12, 1894, 435. ssArchiv. Math. Phys., 14, 1896, 441-3. 
"Math. Quest. Educat. Times, 69, 1898, 99-104. Cf. Meissner,"^ 138-9. 
'"/feid., 87-88, 112-4; 71, 1899, 93-9; Mess. Math., 28, 1898-9, 120-149, 192 (correction). Cf. 
Meissner,"8 137-8. siMathesis, (2), 9, 1899, 241. 
"«Sphinx-Oedipe, 1906-7, [1-2, 17, 33], 49-50, 54, 65-7, 77-8, 81-4; 1907-8, 33-5; 5, 1910, 
145-7; 6, 1911, 157-8. s^Math. Quest. Educat. Times, 71, 1899, 123-4. 
"^BuU. See. Philomathique de Paris, (9), 10, 1908, 10-26. 
s'Assoc. frang., 38, 1909, 145-156; Sphinx-Oedipe, Nancy, 1908-9, 129-134, 145-9; 4, 1909, 
3« Trimestre, 17-25. 
i»»Abh. Bohmischen Gesell. Wiss., Prag, 2, 1786, 140-7. 
