712 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxiv 



three special solutions is 2, 16, 21, 25; 5, 14, 23, 24, given by p = 5 = 3, 

 5 = 2,7 = 5. 



G. Tarry 46 republished his 34 results and noted that 



Ai, - , A k =B 1} , B k , A i +A k . i = 2h = B i +B k - i (i = l, - - -, k) 

 imply AI, - , Ak n = l B\, , B k , as shown by subtracting h from every term 

 of the given equations. A. Aubry concluded that 



A, B, C, -A, -B, -C = A, B f , C f , -A', -B f , -C' 

 if 



A = ab+al3+ba-3al3, B = -ab+ap+ab+3a(3, C =2aj3 -\-2ab, 

 A' = ab+ap-ba+3(xp, B'= -ab+ap-ab-3ap, C' = 2a0-2ab, 



since ZA 2 = ZA' 2 , SA 4 = S^' 4 . Take a = 1, a = 2, 6 = 3, = 4 and add 32 to 

 every term; thus 



1, 12, 21, 43, 52, 63 = 3, 7, 28, 36, 57, 61. 





Aubry noted that AI+X, Bi+y = Ai, B 1} x+y if AiX-\-Biy xy. Hence set 



i = ab, Bi = cd, x = ca, y = ba, a = a-\-d. Thus, if A, J5 = , 77, ", then 

 = ab+bd+cd, B = ab-}-ac-\-cd, whence 



But a?-\-ad+d 2 has besides 3 only prime factors of the form 6/c+l. If 





A 2 -AB+B 2 is divisible by 3, A+B = 3h and A, B = A-h, B-h, 2h. 



2 



Hence A, B = %, 77, f is solvable if and only if A*AB+B 2 is a multiple of 

 3 or has at least two prime factors 6&+1- 



Crussol 47 solved a, b, c, d = a\, bi, Ci, d\. After adding a suitable constant 

 to each term we have a+6+c+d = 0. Set 



A=a+b= c d, Ai 

 = a-b, 2B l = ai-b l , 2C=c-d, 

 Then 



A(B+C}(B-C)=A l (B 1 +C l }(B 1 -C 1 }. 



The general solution of the latter is A = \px, B-\-C = ^qy, B C = vrz, 

 Ai = fj.rx ) Bi-\-Ci = vpy, Bi Ci = \qz. Then the former condition becomes 

 ex 2 =fy <2 +gz 2 , where e = ju 2 r 2 X 2 p 2 , f=fj?q 2 v-p 2 , g = v 2 r~\ 2 q 2 . From the 

 evident solutions (x, y, z) = (v, X, ju) and (q, r, p), we get the general solution 



x = v(a 2 f+Pg), y = X(a 2 /-/3 2 0) +2/iojfy, z = (< x 2 f-pg) -2X/3/. 



L. Bastien 48 proved the impossibility of xi, , x n = yi, - - -, y n when the 

 x's do not form a permutation of the T/'S. For, the elementary symmetric 

 functions of the x's equal those of the y's, so that the x's are the roots of 

 the same equation of degree n as the y's. 



48 Sphinx-Oedipe, num^ro special, June, 1913, 18-23; 1'enseignement math., 16, 1914, 18-27 



(prepared for press by Aubry after Tarry 's death). 

 " Sphinx-Oedipe, 8, 1913, 156-7; special case X=/*=y = l, p. 134. 

 " Ibid., 171-2. 



