316 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ix 



and this number equals 2042 (dg d\\ where d 3 ranges over the divisors 

 4/i + 3 of SM + 6, and di over the divisors 4h + 1- 



In (2) replace rffo) by 27 h (2 2 )0 1 (? 2 ) and 8\(q) by d\(q*} + r?^ 2 ). Then 

 change q z into q. We get 



7h0! + 387?i0? + i?i0! + 20ijX + 20i?J05 = 22(2m + l) 4 g (2w+1) / 4 /(l + g (2m+1) ' 2 ). 

 Equating the coefficients of q N+ *'* and those of q N+l '\ we get 



3) + lOGs, 7 (4JV + 3) = 2(- l)-+ 1 (2m + I) 4 , 

 G 9t i(47V + 1) + 38G 5 , 5 (W + 1) = 22(- l)(2m + I) 4 , 



where 2m + 1 ranges over the odd divisors of 4N + 3 and 4JV + 1, respec- 

 tively. The first formula gives for the total number 120(67, 3 + G St 7 ) of 

 decompositions of 4N + 3 into ten squares the value 122(c?s d\), due 

 to Eisenstein. 2 



For 12 squares, it is shown that 



(2m + 1) 



m=0 



Thus the total number 66(Gio, 2 + 14G ? 6 , e + G z , 10) of decompositions of 

 4N + 2 as a sum of 12 squares equals 2642d 5 , d ranging over the divisors of 

 4N + 2. Changing g into g 2 , we find that 



4) + 4 . 8 (8JW + 4) = 162(2m + I) 5 , 

 summed for the divisors 2m + 1 of 8M + 4. Next, 



gives Gg, 4 (8Af) + G 4 . 8 (8Af) = 162m 5 , m being such that 2M/m is odd. 

 By these and a more complex relation, one may obtain the total number 



of decompositions of 4N into 12 squares, and thus prove Liouville's 9 theorem. 



K. Petr 49 proved Liouville's 12 formula for the number of representations 

 of 2"m as a sum of ten squares by use of the theta functions with the char- 

 acteristics (1, 1), (1, 0), (0, 1), (0, 0) and formulas in Jacobi's Fundamenta 

 Nova (p. 101). Also, Liouville's 9 result on 12 squares by use of the fourth 

 derivatives of P(w). 



E. Dubouis 50 wrote S n for a sum of n squares each > 0. For k > 45, 

 the odd number k 1 or k 4 is a 4 , whence k is a S 5 . The only numbers 

 not 5 's are stated to be A = 0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33. Every 

 number ={= A + 1 is a 6 . The numbers not 6 's are stated to be B = 1, 

 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 19. The -only numbers not a S 6+n are the 

 B + n and the first n integers. 



* J. V. Uspenskij 51 discussed the representation of numbers as sums of 

 squares. 



Archiv Math. Phys., (3), 11, 1907, 83-5. Petr. 43 

 60 L'interm&liaire des math., 18, 1911, 55-56. 

 Math. Soc. Kharkov, (2), 14, 1913, 31-64. 



