CHAP, xxi] EQUATIONS OF DEGREE THREE IN Two UNKNOWNS. 601 



To find two integers the difference of whose squares is a cube and the 

 difference of whose cubes is a square, J. R. Ambler 061 took x 3 +2 and x 3 2 

 as the numbers, the difference of whose squares is (2x) 3 - The difference of 

 the cubes is a square if 3z 6 +4 = D = (2x 3 2) 2 , x = 2. J. Davey used the 

 numbers x, /./ and set x 2 y 2 = z 3 , x+y = n 2 z, which give x, y in terms of z. 

 Then x 3 y 3 = D if 3n 8 -\-z 2 = D = (m 4 z} 2 , which gives z. 



W. Snip 362 made x 2 +y 2 and r 3 +?/ 3 squares by t king x=(m z n 2 )v, 

 y = 2mnv. Then x 3 +y 3 = a 2 b 2 v 2 determines y rationally. 



J. Anderson 363 made x+y a square and x y, x 2J ry 2 cubes by setting 

 xyi=(pqiy. Thenx y = p 3 3p 2 q 3pq 2 -\-q 3 = (q p') 3 tip = 3q. Hence 

 x = ISq 3 , y = 2Qq 3 , x+y= D if 5 = 11. Ashcroft used the numbers (x*x 3 )/2 

 whose sum is z 4 and difference is z 3 . Their sum of squares is (4a; 8 +4.T 6 )/8, 

 which is a cube if 4a; 2 +4 = 5 3 , x = 11/2. 



S. Ward 363 " took y = x+Y, Y = Sr 3 , x=Yz. Then (x 2 +y 2 )/Y 2 equals 

 2z 2 +2z+l, which is the cube of 1+22/3 if z = 9/4. Then x+y = 44r 3 = D 

 if r=ll. 



Several 364 found two integers whose sum is a square and difference a 

 cube, while if each number be doubled the new sum is a cube and difference 

 a square. Take x + y = 4a 6 , x y = 86 6 . 



To make x y, x 2 y 2 , x 3 y* rational squares [Cole 356 ], J. Whitley 365 

 used the numbers x = 2z 2 -{-2v 2 , y = 2z~ 2v 2 ; then shall 



which is true if z = v or z = 2v. H. Godfray took x m 2 -\-n 2 , y = 2mn', then 

 x 2 +xy+y 2 = (m 2 +mn+5n' 2 /2) 2 if n= 4m/7. 



Several 366 solved x+y=n, x 2 +y 2 = D, x 2 +if = x*+y 2 . 



Several 367 found two numbers the difference of whose squares is a cube 

 and difference of cubes a square. 



H. W. Curjel 368 found two numbers x, y whose sum and difference are 

 squares, sum of squares a cube, and ^um of cubes a square. By the first 

 and last conditions, x 2 xy-\-y 2 D, which holds if x = z(2mn n 2 ), 

 y = z(m 2 n 2 '). Then yx are squares if m = 9, n = 4, z= D, whence x = 56z, 

 y = 65z. Then x 2 +y- = 7361s 2 . Thus take z = 7361 4 . 



P. F. Teilhet 369 stated that all pairs of numbers whose sum and sum of 

 squares are squares are (A 2 B 2 )M 2 N and 2ABM 2 N, where A and B are 

 relatively prime and not both even, N=A 2 B 2 +2AB, and where M~N is 

 an integer. He asked when also the sum of their cubes is a square, as for 

 345,184. 



361 Ladies' Diary, 1816, 38-9, Quest. 1291; Leybourn's M. Quest. L. D., 4, 1817, 221-3. 



362 The Gentleman's Math. Companion, London, 4, No. 20, 1817, 659-60. 

 Ibid., 4, No. 21, 1818, 719-21. 



363o Young's Algebra, Amer. ed., 1832, 342-3. 



364 Ladies' Diary, 1821, 32-5, Quest. 1362. 



365 The Lady's and Gentleman's Diary, London, 1849, 49-50, Quest. 1779. 

 ^6 Math. Visitor, 1, 1880, 100-1, 126. 



367 Amer. Math. Monthly, 1, 1894, 95-6, 325. 



368 Math. Quest. Educ. Times, 62, 1895, 51-2. 



369 L'interme'diaire des math., 10, 1903, 124. Cf. papers 139-40 of Ch. XVI. 



