482 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvi 



TWO FUNCTIONS OF ONE UNKNOWN MADE SQUARES. 



C. G. Bachet 126 treated the double equality 



by_factoring the difference N into | and 4JV, where 4JV is the double of 

 VlN 2 , and equating the squares of (4JV=F|) to the given left members, 

 whence, in either case, %N = 65/64. If the second equation is changed to 

 4N 2 -N-1 = D, use the factors 1 and 4#. 



For N 2 -12=D, y.ZV-12=n, use the factors N andJV-13/2 of the 

 difference, so that their sum shall contain the double of V./V 2 . 



For 4AT 2 -]V-4=n, 4N 2 +15N=n, use the factors 4 and 4JV+1 of 

 the difference. 



For N 2 - 6144^+1048576= D, N+64= D, first multiply the latter by 

 16384. 



Fermat 127 treated many double and triple equalities. 



J. L. Lagrange 128 considered, briefly the system 



(1) a+fc#+cz 2 =D, a+/3x+7# 2 =D. 



If a+&/+c/ 2 = gr 2 , the general solution of the first is 



Then the product of the second quadratic by (?7i 2 c) 2 is a quartic function 

 of m. There is no known rule to make the latter a square. If a a = Q, 

 set x = l/y; we are led to the simple problem fo/+c= D, /3?/+7= D. 



R. Adrain 129 treated ax- +6 = D, c# 2 +d= D, given ar 2 +6 = e 2 , by setting 

 x = r+y. Then ax 2 +b = e~+2ary+ay~=(zy e) 2 determines y rationally 

 in z. For this value of y, the second condition becomes Q = D , where Q is 

 a quartic in z; but no treatment is given. Next, consider (1) for the case 

 in which c and 7 are squares ; by multiplication by squares, we may assume 

 that the coefficients of x 2 are equal and proceed as in the following example. 

 For # 2 -#+7 = A 2 , x z -7x+l=B 2 , we have 6#+6 = A 2 - 2 . Take 



2x+2 = A+B, 3 = A-B. 



Inserting a: +5/2 for A in the first given equation, we get # = 1/8. 



Several 130 solved 1-8^=D, x-4x 2 +4=D by inserting # = (l-a 2 )/8 

 into the second condition. Two answers are 



# = 19740/177241, 72165/578888. 



W. Welmin 1300 employed the elliptic function 0(X) obtained by the in- 

 version of the integral 



(2) X-f 



Jo 



126 Diophanti Alexandrini Arith., 1621, 439-440. Comment on Diop. VI, 24 (p. 177 above). 



127 Oeuvres, III, 329-376, French transl. of J. de Billy's Invcntum Novum. See de Billy 65 of 



Ch. IV; Fermat 9 -' 1 and Ozanam 16 of Ch. XV; Fermat 373 of Ch. XXI; Fermat 40 of Ch. 

 XXII. 



128 Additions to Euler's Algebra, 2, 1774, 557-9. Euler's Opera Omnia, (1), I, 596; Oeuvres 



de Lagrange, VII, 115-7. Extracts by Cossali, 37 108-113. 

 139 The Math. Correspondent, New York, 1, 1804, 238-240. 

 130 Math. Miscellany, Flushing, N. Y., 1, 1836, 67-72. 

 1300 Annales Univ. Warsaw, 1913, 1-17 (in Russian). 



