CHAP, xxi] SYSTEMS OF CUBIC EQUATIONS IN n>3 UNKNOWNS. 613 



Thus A 2 -5 2 =(2r+ra)O-p). Take A+B = 2r+m, A-B = n-p, which 

 give A. Hence r 2 +2m-f mn A 2 gives 



_ a?bc | (ab + ac be) 2 



abac bc 



Taking any values of o, 6, c which satisfy a6c+l = D, we get an answer. 

 For example, a = |, 6 = 2, c = 3 give Young's answer = 16016/25. 



On four integers the sum of any two of which is a cube, see Lenhart, 93 etc. 



A. Genocchi 436 noted that early arithmeticians knew that x = 3, y = 4, 

 z = 5, s = 6 satisfy xy = 2s, x z -\-if = z~, ar+2/ 3 +z 3 = s 3 , and proved that this 

 is the only integral solution. If the third condition is replaced by 

 x 3 +y 3 -}-z 3 = s n t, where n>l and t is an unknown integer, he proved that 

 either 6 = 1, a = 3, n = 2, m = l, t = 3 or ra = 3, t = l, or 6 = 1, a = 2, ?i = 3, 

 m = t = l, or 6 = 1, a = 2, n = 2, m = 2, t = 3 or m = l, t = Q. 



P. W. Flood 437 noted that the six cubes 



-_ 



\4J > \3) , \12 , 6J , \168) ) 



of which the sum of the first three is 1/8 and the sum of the last three is 1/8, 

 are such that on adding any one to the square of the sum of the remaining 

 five we obtain a square. 



U. Bini 438 considered xyz = uvw with Sz 2 = Zw 2 or Z# 3 = 1u?, and So: = Zw, 

 Z:r 3 = Zti 3 , the second pair being equivalent to 2z = Za;', xyz x'y'z'. 



L. E. Dickson 439 showed how to obtain all sets of integral solutions of 

 the last pair of equations, as well as of the pair 440 



xyz=x'y'z', xy+xz+yz=x'y'+x'z'+y'z', 



which express the condition that two rectangular parallelepipeds shall have 

 integral edges, equal volumes and equal surfaces. Cf. papers 16-18 of 

 Ch. XVII. 



A. Ge"rardin 441 noted that d 3 x 2 , d?y 2 , d?z 2 , d 3 t z are all squares, 

 where d=x+yz t, if = 65, ?/ = 488, 2 = 481, t1. 



Ge"rardin 442 gave three sets of solutions of x 3 +i/+z* t 3 H-^ 3 + v 3 , xyz = tuv, 

 including the solution 



- 



.) P 

 2 



p pq pq q q p 



The same pair of equations and Sx = 2t have the solution 



x t y u z v 



- = - = pq r~, - = - = gr p-, - = - = pr 



p q q r r p 



436 Atti Accad. Pont. Nuovi Lincei, 19, 1865-6, 49; Annali di Mat., 7, 1865, 157; French 

 transl., Jour, de Math., (2), 11, 1866, 185-7. 



437 Math. Quest. Educ. Times, 70, 1899, 52. 



438 L'interme'diaire des math., 16, 1909, 41-3, 112. Cf. Desboves 302 ; also Sphinx-Oedipe, 



8, 1913, 140, and Ch. XXIV. 



439 Messenger Math., 39, 1909-10, 86-7. 



440 Ibid., and Amer. Math. Monthly, 16, 1909, 107-114. 



441 L'intermediaire des math., 23, 1916, 76. 



442 Nouv. Ann. Math., (4), 15, 1915, 564-6. 



