358 HISTORY OF THE THEORY OP NUMBERS. CHAP, xivj
F, Landry5 used the method of Fennat, eliminating certain squares by their endings and others by the use of moduli.
C. Henry6 stated that Landry 's method is merely a perfection of the method given in the article "nombre premier" in the Dictionnaire des Mathe*matiques of de Montferrier. It is improbable that the latter invented the method (based on the fact that an odd prime is a difference of two squares in a single way), since it was given by Fermat.
F. Thaarup7 gave methods to limit the trials for x in x2—y2~n. We may multiply n by /=a2— 62 and investigate nf=*X2 — Y2, X=ax—by, Y=bx—ay. We may test small values of y, or apply a mechanical test based on the last digit of n.
C. J. Busk8 gave a method essentially that by Fermat. It was put into general algebraic form by W. H. H. Hudson.9 Let N be the given number, n2 the next higher square. Then
where n, r2, . . . are formed from r0 by successive additions of 2n+l, 2n+3, 2n-f5, ____ Thusrm==r0+2?7m+w2. If rm is a square, N is a difference of two squares. A. Cunningham (ibid., p. 559) discussed the conditions under which the method is practical, noting that the labor is prohibitive except in favorable cases such as the examples chosen by Busk.
J. D. Warner9a would make AT= A2— B2 by use of the final two digits.
A. Cunningham10 gave the 22 sets of last two digits of perfect squares, as an aid to expressing a number as a difference of two squares, and described the method of Busk, which is facilitated by a table of squares.
F. W. Lawrence11 extended the method of Busk (practical only when the given odd number N is a product of two nearly equal factors) to the case in which the ratio of the factors is approximately Z/ra, where Z and m are small integers. If I and m are both odd, subtract from ImN in turn the squares of a, a+1, . . ., where a2 just exceeds ImN, and see if any remainder is a perfect square (b2) . If so, ImN = (a+ T}2 — 62.
G. Wertheim12 expressed in general form Fermat's method to factor an odd number m. Let a2 be the largest square <ra and set m = a2+r. If p=2a+l — r is a square (n2), we eliminate r and get ra = (a+l-fn) X (a+1 — n). If p is not a square, add to p enough terms of the arithmetic progression 2a+3, 2a+5, . . , to give a square:
6Aux mathematicians detoutesles parties du monde: communication sur la decomposition des nombres en leura facteurs simples, Paris, 1867. Letter from Landry to C. Henry, Bull. Bibl. Storia Sc. Mat., 13, 1880, 469-70.
6Assoc. fran$. av. so., 1880, 201; Oeuvres de Fennat, 4, 1912, 208; Sphinx-Oedipe, 4, 1909, 3« Trimestre, 17-22. 7Tidsskrift for Mat., (4), 5, 1881, 77-85.
"Nature, 39, 1889, 413-5. 'Nature, 39, 1889, p. 510.
»aProc. Amer. Assoc. Adv. Sc., 39, 1890, 54^7.
"Mess. Math., 20, 1890-1, 37-45. Cf. Meissner,"8 137-8.
"Ibid., 24, 1894-5, 100.
"Zeitschrift Math. Naturw. Unterricht, 27, 1896, 25fr-7.