CHAP, xvii] MISCELLANEOUS SYSTEMS OF Two EQUATIONS. 489 



E. Lionnet 31 stated that 1 and 5 are the only sums of squares of two 

 consecutive integers whose product is a sum of such squares; 1 and 5 are 

 the only primes x, y, each a sum of squares of two consecutive integers, 

 such that x 2 and y 2 are such sums of squares. Similarly, 1, 13 and their 

 biquadrates are sums of squares of consecutive integers. Cf. Lionnet 314 of 

 Ch. XXII. 



MISCELLANEOUS SYSTEMS OF TWO EQUATIONS. 



Bhjiscara 30 of Ch. XII gave a solution of the system x 2 +y 2 +xy = z 2 , 

 0&+2/)^ + l = D. On systems of two equations involving sums of squares, 

 see papers 108, 176 of Ch. VI; 97, 259 of Ch. VII. 



"Umbra" 32 found numbers ax, bx, ex, - whose sum added to or sub- 

 tracted from the sum of their squares gives a square. Set s = a+b+c-\ ---- . 

 Choose a, b, so that the sum of their squares is a square q 2 (by setting 

 g = a+ra and finding a). Hence q 2 x 2 sx are to be squares. Take t = s/q 2 . 

 Then x~tx are to be squares. Determine x by x~+tx = (k x} 2 . Then 

 x 2 -tx= D if W-2kt-P= D = (n-&) 2 , which gives k. 



R. F. Muirhead, 33 to find pairs of quadratic equations x 2 px+q = Q, 

 x z qx+p = Q, all of whose roots are integers ^0, found all integral solutions 

 i=0 of a+j8 = a'j8', a'+0' = a0. Set r=(a-l)(|8-l), r' = (a'-l) (/S'-l), 

 whence r+r' = 2. It is shown that a 4=0. Hence either r = 0, r' = 2, a' = 2, 

 /3' = 3, a = l, |8 = 5, orr = 2, r' = 0, or r = r' = l, a = a f = p = p' = 2. He solved 

 also the pairs 0a = a'f}', /3' a =a&. 



A. Cunningham 34 solved Si = S z = S 3 , where 



by multiplying by 2-10 3 and setting <zy = 10W/+r. Thus 



009022 



a l a 2 = a. 2 a s = a 3 a i . 



But if four integral squares are in A. P., they are known to be equal. 

 M. Rignaux 35 gave integral solutions of the two systems 



xy+zt=D, xz yt=D; xy+zt = xz-yt = u*. 

 A. Boutin 36 proved that x 2 -2y 2 = 1, y 2 -3z z = l imply ?/ 2 = 4. 



31 Nouv. Ann. Math., (2), 20, 1881, 514. 



32 The Gentleman's Math. Companion, London, 4, No. 20, 1817, 673-5. 



33 Math. Quest. Educ. Times, 70, 1899, 84-6. 



34 Ibid., (2), 10, 1906, 29. 



35 L'intermediaire des math., 25, 1918, 113-5. 



36 Ibid., 26, 1919, 123. 



