600 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



and difference x 3 . Choose the squares 



(1-n 2 ) 2 

 = = 



with the sum 1 and set a = ex 2 , b = dx 2 , either of which gives x. 



J. Leslie 355 made x-\-y and x 3 -\-y 3 squares, by division. 



W. Cole 356 made x y, x' 2 y' 2 , x 3 y 3 all squares by taking x y = a 2 , 

 x+y = m~a?, whence x 3 y 5 =H if 3m 4 +l = D, which holds if m = 2. J. 

 Young took m = 2 initially. 



J. Saul 357 made x-\-y = s 2 , x 2 -\-y 2 = v~ and x 3 +?/ 3 a square. By elimination 

 of y from the first two equations, s 4 2s 2 x+2x 2 = v 2 . Let v = s 2 rx. Then 

 z = s 2 (2-2r)/(2-r 2 ). Then z 2 -:n/+2/ 2 = D if r 4 -6r + 14r 2 -12r+4 = D, 

 say (r 2 -3r+5/2) 2 , whence r = 3/4. 



To divide 358 a given square a 2 into two parts such that the difference of 

 their squares and the difference of their cubes are both squares, an anonym- 

 ous solver called b 2 the difference of the parts, whence the difference of 

 their squares is (a&) 2 . The quotient of the difference of their cubes by 6 2 

 is to be a square, whence 3a 4 +6 4 =D. Put a = bx, x = 2z. Then 

 3z 4 +l=49H ----- |-3z 4 is the square of 7-48z/7+12-5l2 2 /49 by choice of z. 



J. Whitley 359 found two positive fractions such that each plus the square 

 of the other is a square, while the difference of their squares or their cubes 

 is a square. Let the fractions be (ld=4v 2 )/8, whose sum and difference are 

 squares. The difference of their cubes is a square if 3 + 16y 4 = D = a 2 . 

 Let v-\ z, %a = l z+2z 2 . Hence z = \ = v. B. Gompertz took 2 = 02 and 

 y = tz as the fractions, where a = (l+Z 2 )/2. Then x 2 2/ 2 =D. Take 

 x+y 2 = p 2 z 2 , y+x 2 = q 2 z 2 . We get two values of z which are equal if 

 as(qa]=t(pt} ) q+a = s(p+t}. These give p and q. Then a; 3 if = (rzY 

 gives z, which equals the earlier value of z if cs (a?s t 2 )(ast) = D, where 

 c = l/(a 3 - 3 ). Take t = 3, s = l. Hence z = 5/32, ?/ = 3/32. 



S. Jones 360 made x-\-y = a?, x 2 -\-y-=n = (bx y)' 2 by choice of x, y. 

 Then z 3 +2/ 3 =D if 6 4 -26 3 +26 2 +26+l = D = (6 2 -6+^) 2 , whence 6=-J. 

 W. Wright took a = 1, proceeded similarly, and found y from 



Then l-3y+3y*= D if m 4 -6m 3 +14?/i 2 -12?/i+4= D = (m~-3m-2) 2 , 

 whence w = 8/3, y = 15/23. 



Lowry 3600 eliminated x = a? y from z 2 +?/ 2 and x* xy+if and equated 

 the resulting expressions to the squares of a? yrfs and a?yel(sw)', the 

 conditions hold if iy=l, 4r = 3s, e = 5s/4. J. Cunliffe took x = R 2 S 2 , 

 y = 2RS, x z -xy+y 2 = (R 2 -RS+S 2 ) 2 , whence 72 = 45; then the desired 

 numbers are a 2 x/(x+y), c 



355 Trans. Roy. Soc. Edinburgh, 2, 1790, 211. 



356 Ladies' Diary, 1787, 36-7, Quest. 853; Leybourn's M. Quest. L. D., 3, 1817, 155-6. 



367 The Gentleman's Diary, or Math. Repository, No. 55, 1795; Davis' ed., 3, 1814, 235. 

 3! >*Ibid., No. 56, 1796; Davis' ed., 3, 1814, 249. 



359 The Gentleman's Math. Companion, London, 2, No. 12, 1809, 169-71. 



360 Ibid., 3, No. 18, 1815, 323-4. 



3600 New Series of Math. Repository (ed., T. Lcybourn), 3, 1814, I, 169-172. 



