604 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XXI 



To find 381 " three integers whose sum and sums by twos are cubes, take 



x+y+z = (b+ri) 3 , x+y = b 3 , z+z = c 3 . 



Then 



7/+2 = 2(6+n) 3 -6 3 -c 3 = (6+2?i) 3 if b= -(c 3 +6w 3 )/(6rc 2 ). 



J. Cunliffe 382 noted that x-\-y z = a?, x-\-z y = b 3 , y+zx = c z imply 

 x -\-y-\-z = a?+b z +c z , which has been made a cube by many writers. 



Several 3820 found three squares in A. P. the sum of whose square roots 

 is a cube by using the know expressions v(2pqq 2= Fp 2 ), v(p 2 -\-q 2 ) for the 

 roots, and equating their sum vk, where k = p z +4pq-{-q 2 , to s 3 & 3 . 



J. Anderson 383 made xyz+1, xy + l,xz+l and yz+1 all squares by equat- 

 ing the last two to (pz I) 2 and (qz I) 2 , whence x = p 2 z 2p, y = q~z 2q. 

 Then the first two will be the squares of l+2pqz and pqz p q if 

 z = 4+2(p+q)/(pq) and p = q+l, respectively. 



To find three numbers whose sum is a cube and the sum of any two a 

 square, J. Foster 384 took x+y = m 2 a 2 , x-\-z = m 2 b 2 , y+z = m 2 c 2 , x-\-y-\-z = d 3 m 3 , 

 whence ?ft=(a 2 +& 2 +c 2 )/(2d 3 ). Many solvers used the numbers 2(x 2 +y 2 

 2 2 ), 2(x 2 -\-z 2 ?/ 2 ), 2(z 2 +y 2 z 2 ), whose sums by twos are squares. To 

 satisfy 2(x 2 +y 2 +z 2 ) = 8p 6 , take y 2 = 4p 6 -n 2 =(cn-2p*) 2 , which determines 

 n, and take z = 2?nn/(m 2 -\-l'), whence n 2 z 2 = D = x 2 . 



W. Lenhart 385 found that the sum, and sum of any two of, 1982015, 

 2759617 and 44286264 are cubes; they are the excesses of 366 3 over the 

 cubes in 



168 3 +359 3 +361 3 = 2-366 3 . 



S. Bills 386 obtained the same result from 



yl _ 1 xj.3 _ 



' 



The root z involves the square root of Qvx 5 3u 4 18t> 2 +9 which is equated to 

 Qv(x+2a)(x a) 2 . For the resulting value of x, 6v(x+2a) = D if 



-3= D = 



whence v = 3/(4a 3 ) . Take a = |. Several writers 387 solved the same problem. 

 To find three integers in arithmetical progression whose common 

 difference is a cube, the sum of any two less the third is a square, and the 

 sum of the roots of the resulting squares is a square, S. Bills 388 took x' 2 y 3 , 

 x 2 , x 2 +if as the numbers. To make x 2 2y 3 squares, take x = uy, whence 

 u 2 d=2?/ are known to be squares if u = t(p'+q^, y = 2pqt 2 (p"-q 2 ). It remains 

 to make 2t(p 2 + 4pg + g 2 ) pq (p 2 - q~] a square, say of 2pq (p 2 + 4pq + q 2 ) (p 2 - q 2 ) r, 

 thus finding t. Other solvers used the numbers x 2:= Fxy-\-y 2 , x 2 +y 2 in A. P. 



ssia N GW s er i e8 o f Math. Repository (ed., T. Leybourn), 2, 1809, I, 31-33. 



382 The Gentleman's Math. Companion, London, 3, No. 14, 1811, 282-3. 

 3820 New Series of Math. Repository (ed., T. Leybourn), 3, 1814, I, 111-5. 



383 The Gentleman's Math. Companion, London, 5, No. 26, 1823, 238-9. 



384 Ladies' Diary, 1826, 35-6, Quest. 1434. 



385 Math. Miscellany, New York, 1, 1836, 123. 



386 Math. Quest. Educ. Times, 12, 1869, 80. 



387 Math. Visitor, 2, 1887, 84-8. 



388 Math. Quest. Educ. Times, 12, 1869, 91-2. 



