CHAP. VII] SUM OF THREE SQUARES. 267 



The case A = l, B = C = Q expresses the square of a SI as a SL The 

 case A = y, B = (3, C = a expresses the cube of a SI as a SI . 



H. S. Monck 48 noted that if a, 6, c are integral edges of a rectangular 

 parallelepiped and the diagonal d is an integer, then a 2 + 6 2 + c 2 = d 2 , 

 and another has the edges a -\- b -\- d, a + c + d, 6-f-c + d and diagonal 

 a -f- b + c + 2d. From a = 1, b = 2, c = 2, d = 3, we get the new 

 edges 2, 3, 6 and diagonal 7. Cf. papers 25-29 of Ch. XIX. 



S. Realis 49 gave a complicated identity 



X 2 + y 2 + z 2 = t 2 + w 2 + v 2 , a = a 2 + /3 2 + 7 2 - S 2 - e 2 , -, 



said to give all solutions of the equation. He gave a similar identity which 

 is said to give all solutions of SO = ffl . Supplementing the theorem that N 

 is a S] if N has no square factor and is of one of the forms 4p + 1, 4p + 2, 

 8p + 3, he stated that TV is the quotient of x 2 + ?/ 2 + z 2 by the factor com- 

 mon to z 2 , ?/ 2 , z 2 , where x, y, z are given above. 



F. Pisani 50 discussed u 2 + (u + I) 2 = (re - I) 2 + x 2 + (x + I) 2 , whence 

 (2w + I) 2 = 6z 2 + 3. Thus 2u + 1 = 3y, 2z 2 - 3?/ 2 = -JL An infini- 

 tude of solutions is found from the continued fraction for A/3/2. 



S. Re*alis 51 expressed as a E] the sum of the three squares of 



2( 2 - 2 - 7 2 + 5 2 ) + 2a(2/3 + 3 7 + 45) 



and two similar expressions. He gave (p. 501) expressions for QP n and 

 18P n as S] if P = a 2 + 6 2 . 



E. Catalan stated and Re"alis 52 proved that every power of 3 is a sum of 

 three squares prime to 3. Realis (p. 75) expressed n 2 (x 2 + y 2 + z 2 ) as a 

 SI when n = a 2 + a& + 6 2 . 



Catalan 53 proved that, if a = b (mod 3), a 2 + 6 2 is a sum of three 

 squares =f= 0; also if a = 6 (mod x -\- y) and 2xy = D. Also that every 

 power of 3 is a sum of three squares prime to 3. He 54 proved that every 

 even power of a 2 + ab + 6 2 is a S] and gave special identities S] SI = SI. 



O. Schier 55 solved x 2 + y 2 + z 2 = it 2 by setting y = x + p, z = x -{- y, 

 u x + d, and taking + 7 = 5. Then 



2z 2 = S 2 - /3 2 - 7 2 , x 2 = 07 = (y - x)(z - x), 



whence x = yz/(y + z). Multiplying the values by y -\- z, we get the 

 identity of Dainelli. 46 



J. Neuberg 56 noted that x z + 2/ 2 + z* = X 2 + F 2 + Z 2 for 



x/o = y/b = z/c = k z + 3, X = a(k 2 - 1) + 2&(& + 1) - 2c(fc - 1), 

 Y and Z being derived from X by permuting o, 6, c cyclically. 



48 Math. Quest. Educ. Times, 29, 1878, 74. 



49 Nouv. Ann. Math., (2), 18, 1879, 505-6. 



50 Nouv. Ann. Math., (2), 19, 1880, 524-6. Same in Zeitschr. Math. Naturw. Unterricht, 12, 



1881, 268. Cf. Lionnet 183 of Ch. XII. 



61 Ibid., (2), 20, 1881, 335-6. 



62 Mathesis, 1, 1881, 73, 87. 



63 Atti Accad. Pont. Nuovi Lincei, 34, 1880-1, 63-4, 135-6. 



"Ibid., 35, 1881-2, 103-114. Extract in Sphinx-Oedipe, 5, 1910, 54-55. 

 65 Sitzungsber. Akad. Wiss. Wien (Math.), 82, II, 1881, 890-1. 

 56 Mathesis, 2, 1882, 116; (4), 4, 1914, 116-7. 



