492 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvm 



Factoring the difference, we set a(c tyn = A B, 2abc -\-bcn = A+B. In- 

 sert the resulting value of A into the equation involving A 2 . We find that 



n = 4:abc (ab + ac be) / { (ac -\- be ab) 2 4a&c 2 } . 

 {(ac+bc-ab} 2 -4:abc 2 } 2 



/v* j^ . . 



Sabc (ac -\-bc- ab) (ac -\-ab- be) (ac ab be) ' 

 The initial squares will be in A. P. if we take 



a = 2rs-r 2 +s 2 , 6 = r 2 +s 2 , c = 2rs+r 2 -s 2 ; 



whence a = l, 6 = 5, c = 7 if r = 2, 8 = 1. Then a; = 151321/7863240, a result 

 found by J. D. Williams 6 by starting with the squares x 2 , 25x 2 , 49z 2 . For 

 r = 4, s = 3, we get a = 17, 6 = 25, c = 31, x= X, where 



X= (864571) 2 /11011044931800, 



and hence a solution of a 2 X 2 aX=D, , c 2 X 2 c_X"=D [Perkins 28 of 

 Ch. XIV]. 



Hart 7 made k 2 x 2 -{-kx=n for k = a', 6', . Divide by k 2 and set 

 a = l/a', . Then x 2 -\-ax, x 2 -}-bx, are to be squares. Set x=z 2 . 

 Then 2 2 +a, 2~+&, are to be squares. Suppose that 2 2 is a sum of two 

 squares in the required number of ways: Z 2 = m 2 +n 2 = p 2 +q 2 = , and 

 set a = 2mn, b = 2pq, . Then z 2 +a=(m-\-ri) 2 , z 2 -}-b = (p-\-q) 2 , . 



J. Matteson 8 gave the solutions by Hart 5 - 7 with amplifications. 



G. B. M. Zerr 9 solved the system x 2 +y z = z 2 +w 2 = D, z 2 -w 2 =z 2 -?/ 2 = D, 

 also the system 



(m 2 +n 2 ) 2 a; 2 (m 2 +n 2 )a:= D, (m 2 n 2 ) 2 x 2 (m 2 ri r )x= D, 



4m 2 ?i 2 a; 2 zb2mna;= D. 



P. von Schaewen 10 made 4o: 2 2x, 4:X 2 -\-3x, 4 2 +5:r all squares. Setting 

 a; = l/(4a;i), we are to make 1 2xi, l+3#i, l+5^i all squares [von 

 Schaewen 81 of Ch. XV]. 



On three squares which increased or decreased by their roots give 

 squares, see papers 12, 12a, 21, 26, 52-54 of Ch. XIV. For two squares, 

 papers 3, 19 of Ch. XVI; 32 of Ch. XVII. 



THREE LINEAR AND QUADRATIC FUNCTIONS OF TWO UNKNOWNS MADE 



SQUARES. 



Brahmegupta 2 of Ch. XV made x-\-y, xy and xy+1 all squares. 



To find two numbers whose product is a square and product plus the 

 square of either is a square, J. Hampson 11 took b 2 a and a as the numbers. 

 It remains to make 6 2 +l = D (b c) 2 , say, which gives b. R. Mallock 



6 Algebra, Boston, 1840, 413. 



7 Math. Quest. Educ. Times, 39, 1883, 47-9. 



8 Collection of Diophantine Problems with Solutions (cd., A. Martin), Washington, D. C., 



1888, pp. 10-20. 



9 Amer. Math. Monthly, 15, 1908, 17-18. Erroneous solution in J. D. Williams' Algebra, 



1832, 419. 



10 Archiv Math. Phys., (3), 17, 1911, 249-250. 



11 Ladies Diary, 1763, p. 34, Quest. 491; Leybourn's Math. Quest. L. D., 2, 1817, 209. 



