358 History of the Theory of Numbers. Chap, xivi 
F. Landr>'* used the method of Ferinat, eliminating certain squares by 
their endings and others by the use of moduU. 
C. Henry^ stated that Landry's method is merely a perfection of the 
method given in the article "nombre premier" in the Dictionnaire des 
Math^matiques of de Montferrier. It is improbable that the latter in- 
vented 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. Thaarup^ gave methods to limit the trials for x in x'^ — y^ = n. We 
may multiply n by f = a^ — b^ and investigate nf = X" — Y^, 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. Busk^ gave a method essentially that by Fermat. It was put 
into general algebraic form by W. H. H. Hudson.^ Let N be the given 
number, n" the next higher square. Then 
N=^n'-ro={n+iy-r,= ..., 
where ri, r^,... are formed from ro by successive additions of 2?i + l, 2nH-3, 
2n+5, . . .. Thus r^ = ro+27?27i+??r. If r^ is a square, iV is a difference 
of two squares. A. Cunningham {ibid., p. 559) discussed the conditions 
under w^hich the method is practical, noting that the labor is prohibitive 
except in favorable cases such as the examples chosen by Busk. 
J. D. Warner^*" would make N = A~—B'^ by use of the final two digits. 
A. Cunningham^^ 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. Lawrence^ ^ extended the method of Busk (practical only when 
the given odd number iV is a product of two nearly equal factors) to the 
case in which the ratio of the factors is approximately l/m, where I and m 
are small integers. If I and ??? are both odd, subtract from bnN in turn the 
squares of a, a+1, . . . , where o^ just exceeds ImN, and see if any remainder 
is a perfect square (6") . If so, ImN = (a+ T)'^ — 6^. 
G. Wertheim^^ expressed in general form Fermat's method to factor 
an odd number ?n. Let a^ be the largest square <m and set m = a~+r. 
If p=2a+l— r is a square (n^), we eliminate r and get m = {a-{-l-\-n) 
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: 
p+(2a+3)-}-...H-(2a+2n-l)=s". 
'Aux math^maticiens de toutes les parties du monde: communication sur la decomposition des 
nombres en leurs facteurs simples, Paris, 1867. Letter from Landry to C. Henry, Bull. 
Bibl. Storia Sc. Mat.. 13, 1880, 469-70. 
•Assoc, frang. av. sc, 1880, 201; Oeuvres de Fermat, 4, 1912, 208; Sphinx-Oedipe, 4, 1909, 3« 
Trimestre, 17-22. ^Tidsskrift for Mat., (4), 5, 1881, 77-85. 
•Nature, 39, 1889, 413-5. 'Nature, 39, 1889, p. 510. 
»'Proc. Amer. Assoc. Adv. Sc, 39, 1890, 54r-7. 
"Mess. Math., 20, 1890-1, 37-45. Cf. Meissner.i" 137-8. 
"/Wd., 24, 1894-5, 100. 
»»Zeit8chrift Math. Naturw. Unterricht, 27, 1896, 256-7. 
