CHAP, xix] PROBLEMS RELATED TO THE LAST ONE. 515 



Set -ABC = mr+n+t(nr-m). Then, for d = n(P-l)+2mt, 



To obtain integral solutions, set 5 = (p 2 +l)/A, C=(q 2 +l)/A. Then 



is a square if A = n = p q. Then B A-\-C+2q. It remains to make 

 <? 2 +l divisible by A, which requires that A = IZ\. If A = 5, # = 5w2, then 

 C = 5w 2 4t/H-l, J3 = 5u 2 +14w + 10 or 5^+6^+2. Among other ways of 

 obtaining integral solutions, take AB = 1+p 2 , (AC-lXfiC-l) = (wC+1) 2 , 

 whence (7 = (A+5+2m)/Q, where Q = AB-m 2 . Then 



Hence we set Q = n 2 , whence ra 2 +n 2 = jp 2 +l. Take m = ap-}-cx, n = ap a, 

 where a 2 + 2 = l; for example, = (/ 2 -0 2 )/(/ 2 +0 2 ), a = 2fg/(f 2 +g 2 ). Then 



For/=2p, = 1, C 



where/i = 4p 2 +l. Next, we take/=/i, g = 2p. In this way Euler obtained 

 C=(A+B)M' 2 2pMN, where (M, N) = (l, 1), (4p 2 +l, 4p 2 +3),--- are 

 given by a recurring series with the scale of relation 4p 2 -f2, 1; he gave 

 the general terms. 



J. Leslie 81 made xy-\-l, #2+1, 2/2 + 1 squares by factoring (cf. Buchner 83 ). 



P. Cossali 82 gave the result due to Saunderson. 78 



Fr. Buchner 83 treated xy+1 =p 2 , xz+1 = q-, yz+1 =r~. Then 



p+1 q+1 lf ^ r+1 



- 



m n 

 Thus p, q, r and hence also x, y, z are functions of m, n, I. 



A. B. Evans, 84 to make xy 1, etc., squares, took x = a?-\-b 2 , y = c 2 -{-d 2 , 

 z = e 2 +f 2 , E = bc-ad, F = be-af, G = de-cf. Then xy-E*, xz-F 2 , yz-G 2 

 are squares. Take e = a+c, f=b+d. Then F = E, G=E. It remains 

 only to make E=l. 



E. Bahier 85 noted the answer a 1, a, 4a 1 and gave de Sluse's 77 values 

 with z = l and Saunderson 's 78 with the second z. 



PROBLEMS RELATED TO THE LAST ONE. 



Diophantus, III, 17, 18 [19], treated the problem (which evidently 

 reduces to the last one): to find three numbers such that the product of 

 any two increased [diminished] by the sum of those two gives a square. 86 



81 Trans. Roy. Soc. Edinb., 2, 1790, 209, Prob. XII. 



82 Origine, Trasporto in Italia . . . Algebra, 1, 1797, 102. 



83 Beitrag zur Auflos. unbest. Aufg. 2 Gr., Prog. Elbing, 1838, p. 9. 



84 Math. Quest. Educ. Times, 14, 1871, 75-6; 29, 1878, 90-1. 



86 Recherche M<thodique et Proprietes des Triangles Rectangles en Nombres Entiers, Paris, 



1916, 198-9. 

 86 In Diophantus IV, 38, 40, the results are to be given numbers, instead of squares. His 



condition that each number must be 1 less than a square is not necessary, as noted by 



Stevin, Les Oeuvres math, de Simon Stevin . . . par A. Girard, 1625, 589; 1634, 148. 



Thus if the numbers are 14, 23, 39, an answer is 4, 2, 7. 



