ON THE THEORY OP NUMBERS. 339 



[4N]=S[2N l = (2^ + l) 2 + (2^ + l) 2 ]x[2N 2 =(2.r+l) 2 + (2y + l)'- ! ], 

 in which the summation extends of every pair of uneven numbers N\ and N 

 which satisfy the equation 2^=^ + 1^, and the square brackets represent 

 the number of solutions in positive integers of the equations included in them. 

 Observing that [2N l =(2.r + l) 2 4-(2y + l) 2 ] is the excess of the number of 

 divisors of N\ which are of the form 4£ + l, above the number of its divisors 

 which are of the form 4k— 1, retaining the signification of a, ft, a, ft', and 

 denoting by a and b any positive uneven numbers, Ave may transform the ex- 

 pression of [4N] into the following, 



[4N]=[2N-=(rt + 6) a ] + [2N=«a + J/3]-[2N=« a ' + &/3'], 

 in which the square brackets still retain the same signification. Supposing, 

 as before, ftxx., and /3=a + 4«, we have 



[2N=«a + 6/3]=2[2N=a(rt + &)+4n&]; 

 or, putting a =v + 4hi, v being less than 4w, 



[2N==aa + 6/3]=2[2N=r(rt + 6) + 4n(^+6Z,- + 6)]=2[N=Kr+2ny], 

 y being uneven and y-<2n. Again, if in [21$=aa.' + bfi''] we write 4n for 

 a 4- ft', and suppose a>b (the supposition a =b is inadmissible as it would 

 render N even), we have 



[2N=aa' + 6/3']=2[N=a'^+2n&]=2[N=Kff+2ny], 



as before. Hence {28=aa+bpl— [2N=aa'+&/3']=0, and [4N]=[2N 

 = (a + b)oc], i. e. [4N] is the sum of the divisors of N. In this arithmetical 

 process wo determine the coefficient of q™ in P, Q, R, instead of determining 



those functions themselves ; and as the difference Q— R= — S„ " 1 is an 



1 — q* n 

 even function in the analytical process, so the difference [2N"=«a + 6/3] 

 — [2N=«a' + &/3'] vanishes in the arithmetical one. 



Lejeune Dirichlet, in a letter addressed to M. Liouville (Liouville's Journal, 

 New Series, vol. i. p. 210), has put Jacobi's demonstration into a form in which 

 it is more easily followed, but is a little further removed from the analysis. 

 He shows that to every solution of the equation aa + bft=2~N, in which 

 a>/3, there corresponds a solution of the equation a'a.' + b'0'=2'N, in which 

 a.'>ft', and vice versa, the two solutions being connected by the relation 



a!, b' 

 P, -a' 



x + \,x + 2 I 



cc ,x+i x 



b, a 



a, -ft 



where x is the integral number immediately inferior to -i- — , or, which is the 



a — ft 



same thing, to -fi-g,* Hence, as before, [2N=cra + 6/3]=[2N=a'a' + b'ft'], 



iX ~~ fj 



and [4N] is equal to the sum of the divisors of TS. 



128. Theorems of Jacobi on Simultaneous Quadratic Forms. — In an elabo- 

 rate memoir " On Series whose Exponents are of two Quadratic Porms " *, 

 Jacobi has established a great number of elliptic formulae, which are the ana- 

 lytical expression of theorems relating to the representation of numbers by 

 certain quadratic forms. A comparison of the two criteria of Gauss for the 

 biquadratic character of 2 with respect to a prime p of the linear form 8& + 1, 

 leads to a result which will serve as an example of these theorems. By the 

 first criterion, 2 is or is not a biquadratic residue of a prime p of the form 

 8&+1 according as a is even or uneven in the equation p= (4a + l) 2 + 86 2 ; 



* Crelle's Journal, vol, xxxvii. p. 61 and 221 ; or Mathematische Werke, vol. ii. p. 67. 



2 a2 



