248 report — 1859. 



Conversely, the circumstance that a trigonometrical summation should depend 

 on the quadratic characters of integral numbers, may serve of itself to show 

 the use of abstract arithmetical speculations in other parts of analysis. 



21. Gauss s Sixth Demonstration. — This demonstration depends on an 

 investigation of certain properties of the algebraical function 



s=p— 2 



s=0, 



in which p is a prime number, y a primitive root of p, k any number prime 

 to p, and x an absolutely indeterminate symbol. These properties are as 



follows : — 



p-\ 1—xf 



(1) W - (-1) 2 p is divisible by j— — , 



(2) Ik— f-Ui > s divisible by 1— xP, 



(3) If k=q be a prime number, 

 £ ; ?— £ 3 is divisible by £. 



From (l)we may infer that '£ x «-' — ( — 1)4 0»-iH?-i)/? T" is divisible by 

 ; and, by combining this inference with (1) and (2), we may conclude 



1—xP 

 1-x 



\—xp 



that £ i a i «-5,)-(-l)Ti' [(-l^-^'-^V-^)] 

 is also divisible by ■ ; that is to say, 



J ~ ~~ 3C 

 (-1) 2 />[(-l) i> 2 -(Jj 



is the remainder left in the division of the function £, (£^— £ g ) by ^— 



But every term in that function is divisible by q; the remainder is therefore 

 itself divisible by q. We thus obtain the congruence 



(_l)H/>-i)(?-i)joV~ = (^\mod?, 



which involves the equation /^U-iW( — 1)* (/>-D te-D. 



Gauss has given a purely algebraical proof of the theorems (1), (2), and 

 (3), on which this demonstration depends. The third is a simple consequence 

 of the arithmetical property of the multinomial coefficient, already referred 

 to in Art. 10 of this Report; to establish the first two, it is sufficient to ob- 



serve that l k 2 — ( — 1 ) 1~ p, and £* — (-)£> vanish, the first, if x be any ima- 

 ginary root, the second, if x be any root whatever, of the equation xP— 1 =0. 



If, for example, in the function & we put ;r=r=cos \-i sin — ,weob- 



p p 



tain the function \p (k, p), which satisfies, as we have seen, the two equations 

 [i//(A,/>)] 2 =( — 1) 2 p, and ^(k>p)—\- \^{\,p). It is, indeed, simplest 



