ON THE THEORY OF NUMBERS. 341 



does not differ from that subsequently given by Gauss (Theor. Res. Biq. Coram, 

 prima, art. 13), and then deduces the second criterion, as follows, from the 

 first. Since p = (4« + l) 2 + 86 2 = (4a +l) 2 + 16/3 2 , we have [4(«+/3) + l]x 

 [4(«— ft) + l] = (4a-f-l) 2 — 8b 2 . No common divisor of 4a-f-l and b can also 

 be a common divisor of 4(«+/3) + l and 4(«— /3) + l, i. e. of 4a -f 1 and b ; 

 for p is not divisible by any square. The greatest common divisor of (4a+ 1) 2 

 and b 2 must therefore be a product of two relatively prime uneven squares 



c 2 and o' 2 , dividing 4 (a 4- ft) 4-1 and 4(a—ft) + l respectively; 4 («+fi) + l 



is thus a divisor of the quadratic form x 2 — 8>/ 2 , in which as and y are rela- 

 tively prime ; it is, consequently, itself of that quadratic form, and 

 4(«+/3) + 1 = 1, mod 8 ; this congruence implies that a 4- ft = 0, mod 2, or, 

 which comes to the same thing, that 6 = <z+/3, mod 2. It will be seen that 

 this demonstration of the congruence b = a + ft, mod 2, applies to any two 

 representations of any number P by the forms /=(4a + l) 2 + 86 2 and 

 0=(4a + l) 2 + 16/3 2 , provided that in the two representations the four num- 

 bers 4a -J- 1, 4a + i, b, ft have no common divisor. To prove, for every uneven 

 value of P, the truth of Jacobi's equation 2( — 1)"=2( — l) a+ ^, we observe, 

 first of all, that the equation is evidently true if P is not = 1, mod 8, or if P 

 contain an uneven power of a prime of the linear form 8k + 7 ; for in these 

 cases there are no representations of P by either form. We may therefore 

 suppose that P is of the linear form 81- + 1 ; then the equation is true if P 

 contains an uneven power of any prime p of either of the linear forms 81- + 3 ; 

 thus if P=p-" +1 P', where P' is prime to p, and p = P'=3, mod 8, there 

 are no representations of P by <p, so that S(— l) a+/3 =0; let the equations 

 2)' 2 "+ 1 —,v 2 + 2y-, P'=X 2 + 2Y 2 denote generally those representations oi p' iv + l 

 and P' by the form (1, 0, 2), in which the first indeterminate is = 1, mod 4; 

 then the rcpi-esentations of P^) 2 '^ 1 x P' by/ will bo comprised in the for- 

 mula P= (2y Y - .-rX) 2 + 8 ( ,jX + xY \ *; but of the two numbers ^yX+aY), 



|(,yX — *Y) (both being values of the second indeterminate), one is uneven 

 the other even ; whence 2( — 1)*=0=2(— l) a+ ^. Similarly if P^^+'P', 

 where P' is prime top, and^> = P'=5, mod 8, there are no rejtresentations 

 of P by /, and it may be shown that 2( — l) a+/3 = 0=2(-l)*. We may 

 therefore confine ourselves to the case in which P is composed of any powers 

 of primes of the linear form 8£ + 1, and of even powers of primes of the forms 

 8&+3, 5, 7. If, on this supposition, P=P' x P", where P' and P" are rela- 

 tively prime, and each is = 1, mod 8, the sums 2( — 1)° and S(— l) a+p , rela- 

 tive to P, are the products of the corresponding sums relative to P' and P". 

 This may be proved by observing that the representations of P by /[or 0] 

 may be obtained by compounding the representations of P' and P" by that 

 form, and that each representation of P has the character of an even or uneven 

 b [or a+/3] according as the representations of P' and P" of which it is com- 

 pounded agree or differ in respect of that character. Thus it is sufficient to 

 consider the four cases in which (1) P=p", p = 1, mod 8 ; (2) 'B=p 3v , p = 3, 

 mod 8 ; (3) Y=p*>, p = 5, mod 8 ; (4) Y=p*", p = 7, mod 8. In the last 

 of these cases it is evident that 2( — 1)*= + 1=2(— l) a+ ' 3 ; in the others, 

 the proof is supplied by Dirichlet's method, (i) If P —p v , p = l, mod 8, there 

 are two primitive and v — 1 derived representations of P by each form; and 



* For 2yY— jbX=1, mod 4 ; and the representations comprised in the formula are all 

 different, their number being equal to the number of sets of representations of P by (1, 0, 8). 



