CHAP. IX] RELATIONS BETWEEN SQUARES. 323 



and obtained 102 ten decompositions of 266 2 into nine squares by multiplying, 

 two by two, five decompositions of 266 as a sum of three squares. 



E. Barbette 103 used the method of Martin 87 to find squares whose sum 

 is a square. He gave (pp. 87, 96) many sets of consecutive squareswhose 

 sum is a square, [cf. Lucas 70 ] 



E. Miot 104 stated that, if 2 k < m ^ 2 fc+1 , the square of a sum of m squares 

 is a sum of 2 k + 1 squares. 



E. N. Barisien 105 noted that the sum of the squares of z 6 , 4sV> xy 5 , 

 2y* and 2xy(2x* + 5x*y z -f 2i/ 4 ) equals the square of x 6 + Sx 4 y 2 + Sx 2 y 4 + 2t/ 6 , 

 and gave seven squares whose sum is a square. 



L. E. Dickson 106 gave a history of the problem to express the product of 

 two sums of n squares as a sum of n squares. 



On I 2 + + x 2 = % 2 , see Lucas 151 of Ch. I. On x\ + + xl = R 2 , 

 see Turriere 115 of Ch. VII, Escott 261 of Ch. XXI and paper 94, p. 322. On 



xl+ h xl = y p , see papers 96a, 98 of Ch. XX ; 268 of Ch. XXI (p = 3) ; 



and papers near the end of Ch. XXII (p = 4). By Landau 21 of Ch, XXV, 

 every definite polynomial in x is a sum of the squares of 8 polynomials. 



102 Mathesis, 10, 1910, 185. 



103 Lea sommes de p-i&mes puissances distinctes e"gales a une p-ime puissance, Liege, 1910, 



77-104. 



104 L'interme'diaire des math., 19, 1912, 195. 



105 Sphinx-Oedipe, 8, 1913, 142. 



106 Annals of Math., (2), 20, 1919, 155-171, 297. 



