504 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. XIX 



S which is a sum of two squares in four ways, employ 



Hence S = A 2 +a 2 = 2 +/3 2 = C 2 +7 2 = > 2 +5 2 if 



A = eE + -fF+, B = eF. -/EL, C = eE. -/*L, D = eE + +/F+, 

 a = cF + +fE+, /3 = e^_+/F_, T = eF_+/#_, d = eF+-fE+. 



It remains to satisfy the condition S= SA 2 or, if we prefer, A 2 +D 2 = 7 2 B 2 , 

 viz., 



Divide by / 2 and set . =fw. Thus 

 w; 2 (ac+6rf) 2 +(ar/- 

 The roots w are rational if the discriminant 



is a square. Take a = mb, c = nd, mn = r(n V). Then shall 



r(n+l)(m+w r)(rn n r) = D. 



Take n = 2r. Then shall 2r 2 - 3r = D , as is the case for r = 3s 2 / (2s 2 - 1) . For 

 s = l, we get Euler's solution 168, 105, 280, 60. Removing the restriction 

 n = 2r, let (nr+n r)k = (nrn r)l. Then shall n()i+l)(n l)e= D, 

 e = (l+k)l(lk'). Take n = e+x. There results the answer a = 



B. Gompertz 34 employed x 2 , ?/ 2 , s 2 , w 2 , 



Then a; 2 +2/ 2 +2 2 and w 2 -\-y 2 -t-z 2 are squares. Also, x 2 +w 2 +z 2 and 

 are squares if 



for J = ?/ and j = 2. Take p = (g 2 -r 2 )/(2r), ?/=(g 2 +r 2 )/(4r). Then 



and/,= D. Set z = ty, pq/y 2 = b. Then/,= D if (l+ 2 ) 2 +6 2 (l-^ 2 ) = D. 

 Set < = 1+y. The condition becomes z; 4 H ---- = D = (2+Avw 2 ) 2 and holds 

 if A=2-6 2 /2, v=6 2 /4-l. For g = 2, r=l, we get Scott's 38 solution. 



C. Gill 35 treated the problem to find n squares the sum of any n 1 of 

 which is a square. He 36 gave elsewhere his solution for n = 5 and remarked 

 that the smallest numbers given by his formulas are so very large as to 

 discourage any attempt to compute them. For n = 3, see Gill. 13 The 

 method was adapted to the case n = 4 by S. Bills. 37 If z 2 , if, x 2 , w 2 are the 

 required squares, their sum shall equal 



34 The Gentleman's Math. Companion, 2, No. 12, 1809, 182-4. Reproduced (essentially) by 



A. Martin, Math. Mag., 2, 1898, 216. 

 Application of the angular analysis . . . , New York, 1848, 69-76. 



36 The Lady's and Gentleman's Diary, London, 1850, 53-5, Quest. 1797. 



37 Math. Quest. Educ. Times, 16, 1872, 108-110. 



