244 



HISTORY OF THE THEORY OF NUMBERS. 



[CHAP, vi 



sets of solutions x z , y 2 of N 2 = x\ + yl = (xj +2/i)(e 2 + f 2 ). Then 

 NS = xj + 2/3 has 8 sets, etc. It is proved (pp. 503-6) that if p and p' 

 are primes 4e 1, no one of p, p' or pp f is a EL 



V. Schlegel" stated that the numbers (8X + 7)4 M are the only ones 

 not a sum of fewer than four squares; the numbers (4X + 3)2" and the 

 products of two relatively prime numbers of that form are the only numbers 

 not a sum of fewer than three squares. The numbers representable as a El 

 are s-2", where s = 4(X 2 + v z + v) + 1. The numbers representable in n 

 ways as a El are 2" tunes the product of n factors s. 



T. Muir 100 noted that by Lagrange's theorem any integer A is of the form 

 x 2 + y z if in the continued fraction for VA the period of the partial de- 

 nominators has an odd number of terms. Muir 101 gave formulas for x 

 and y. For, the general expression for such an integer is A = R z + S, 



For 



R = 



S = 



where ai 2 - a n a n 

 example, 



K(a 2 - - - a 2 ) 2 , 

 a 2 i is the period, while K is a continuant. 



1 

 

 



Then A = re 2 + y 2 , 

 2x = {^(G! a n )' 



a n ) 



y = 



o n _i) }M 



a n ) 3 . 



When M = K(di a 2 ), A = x 2 + y 2 is also the sum of 3 squares. 



E. Lucas 102 gave the complete solution of u z + v 2 = y* and stated that 

 the same process applies to u 2 + v 2 = y z ". 



S. Roberts 103 derived all the decompositions into the sum of two squares 

 of an odd positive integer D, containing no square factor, and such that 

 t z Du 2 = 1 is solvable in integers, by developing into a continued 

 fraction V/V/M, where M and N are complementary factors of D and 

 M < VZ>. For D odd, we take M < VD/2. 



G. H. Halphen 104 considered the sum s(x) of the positive divisors d of a 

 positive integer x such that x/d is odd. Then 



|s(x) = s(x - 1) - - s(x 4) + s(x - - 9) d= s(x - - n 2 ) + -, 



99 Zeitschrift Math. Phys., 21, 1876, 79-80. 



100 Proc. London Math. Soc., 8, 1876-7, 215-9. The Expression of a Quadratic Surd as a 



Continued Fraction, Glasgow, 1874, 51. Euler 72 of Ch. XII wrote (a, 6) for K(a, b). 



101 Proc. Roy. Soc. Edinb., 1873-4, 234. 



102 Bull. Bibl. Storia Sc. Mat. Fis., 10, 1877, 243. Cf. J. Bertrand, Traite" &6m. d'algebre, 



Paris, 1850, 244; 1851, 224. Cf. Lucas" of Ch. XXII. 



103 Proc. London Math. Soc., 9, 1877-8, 187-196. 



. 1M Bull. Soc. Math. France, 6, 1877-8, 119-120, 179-180. 



