CHAP. XVI] X z + lf 1 AND X 1 y* 1 BOTH SQUARES. 479 



X- + y~ 1 AND X 2 ?/ 2 1 BOTH SQUARES. 



BMscara 107 (born 1114) gave sets of values of x, y for which x~y---l 

 are both squares : 



_ 1 



Bhdscara 108 treated this problem and the similar one on z 2 2/ 2 +l, as 

 being due to an ancient author. To find two squares whose sum and 

 difference increased by unity are squares, call the desired squares [j/ 2 =]4fc 2 

 and Qc 2 =]5fc 2 1, the latter being a square for k = l or 37. For decrease 

 by unity, use 4/b 2 and 5/c 2 +l, a square for k = 4 or 72. 



Having chosen the coefficient 4, the other coefficient (5) is to be deter- 

 mined so that when 4 is added or subtracted we get a square. Thus 2-4 

 is the difference of two squares. Taking 2 as the difference of their roots, 

 we get the roots to be 1 and 3, whence 5 = 4+l 2 = 3 2 4. Similarly, taking 

 36 as the first coefficient, we must make 72 a difference of two squares. 

 Taking 6 as the difference of their roots, we get 45 as the second coefficient ; 

 taking 4, we get 85. 



J. Cunliffe 108 " solved z 2 =J(c 2 +d 2 )+l, y ^ = i(^-d^ by taking c = d+n, 

 y = r n t whence n = 2dj (2r 2 1 ) by the second condition . Take d = s(2r 2 1 ) , 

 x = ts+l. The first condition is satisfied if (4r 4 +l-J 2 )s = 2/. For t = 2r 2 , 

 we get Bhascara's final answer. 



E. Clere 109 treated the same pair x 2 +y~ 1=2 2 , x 2 y 2 l = it 2 . By 

 subtraction, 2y 2 = z 2 u~. Let y = pq and set z+u = 2q~, zu = p" [thus 

 limiting to integral solutions]. Substituting the resulting values of z, u 

 into the proposed first equation, we get 4o; 2 = 4+4g 4 +p 4 , which is a square 

 if p = q*. Thus we have the special solution 



A. Genocchi 110 proved that all rational solutions are given by 



_2gpq _l+r # 2 (p 4 +4g 4 ) r 



y= I ' I ' 2r 2' 



where p, q are relatively prime integers, q odd; r an integral divisor of 

 2 (p 4 +4g 4 ) and r = g (mod 2). We may give any rational values to g, p, 

 q, r and, without loss of generality, replace r by gr. Then y = pqrld, 

 x=(p*+4q*+r z )/d, where d = p 4 +4g 4 -r 2 . If we set r = 2^ 2 , p = l, we get 

 2/ = 8g 3 , x = 8g 4 +l; if we set r = p z -2q 2 , p= --1/2, we get also the first set 

 by Bhdscara. _ 



107 Lildvati (arith.), 59-61. Algebra, with arith. and mensuration, from the Sanscrit of 



Brahmegupta and Bhdscara, transl. by Colebrooke, London, 1817, p. 27. Lilawati or 

 a treatise on arith. and geom. by Bhascara Acharya, transl. by John Taylor, Bombay, 

 1816, 35. 



108 Vija-ganita (algebra), 194; Colebrooke, 107 pp. 257-9. 



iosa N. ew genes Math. Repository (ed., T. Leybourn), 2, 1809, I, 199. 



109 Nouv. Ann. Math., 9, 1850, 116-8. 



110 Ibid., 10, 1851, 80-85. 



