CHAP. xiV] METHODS OF FACTORING. 367
-AO = 10:r2-12x-4, whence 12z+4 is divisible by 10, so that z = 5d— 2. Then A = 25d2 —26d+6. Thus if we delete the last two digits 7 , 6 of squares -476, we obtain numbers A whose values for d=T9 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 Ami = (lOx - 1) (Wy - 1) . Then
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. Speckmann87 noted that the two factors of JV=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 Hand 77, etc.
G. Speckmann88 wrote the given number prime to 3 in the form 9a+& (fc<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. Christie89 and D. Biddle90 made an extensive use of terminal digits.
E. Barbette91 noted that Wd+u has a divisor 10m — 1 if and only if d+mu has that divisor. Set d+mu = n(Wm— 1), d = 10d'+w'. Then mn^d'+x, Wx=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 10m+l, 10m=»=3.
A. Ge*rardin910 developed Barbette's91 method.
R. Rawson92 found Fermat's1 factors of a number proposed by Mersenne by writing it to the base 100 and expressing it as (a-102+23)(&-102+3).
J. Deschamps93 would use the final digits and auxiliary tables.
A. Ge*rardin94 would factor N (prime to 2, 3, 5) by use of
and a table showing, for each of the 32 values of K< 120, the 16 pairs a, b (each< 120) such that ab^K (mod 120). He factored Mersenne's number.1
FACTORING BY CONTINUED FRACTIONS OR PELL EQUATIONS. Franz von Schaffgotsch100 would factor a by solving az2+l =x2 (having
87Archiv Math. Phys., (2), 12, 1894, 435. 88Archiv. Math. Phys., 14, 1896, 441-3.
"Math. Quest. Educat. Times, 69, 1898, 99-104. Cf. Meissner,1'8 138-9.
90Ibid., 87-88, 112-4; 71, 1899, 93-9; Mess. Math., 28, 1898-9, 120-149, 192 (correction). Cf.
Meissner,138 137-8. «Mathesis, (2), 9, 1899, 241.
fllaSphinx-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. wMath. Quest. Educat. Times, 71, 1899, 123-4.
03Bull. Soc. Philomathique de Paris, (9), 10, 1908, 10-26. °*Assoc. frang., 38, 1909, 145-156; Sphinx-Oedipe, Nancy, 1908-9, 129-134, 145-9; 4, 1909,
3C Trimestre, 17-25. 100Abh. Bohmischen Gesell. Wise., Prag, 2, 1786, 140-7.