ON THE THEORY OF NUMBERS. 773 
r s= ore ree 
F(w)== o av", w representing a root of the equation Pet =0, and y a 
s=0 a 
F (a=*) F (eo) 
F (0-*") 
is an integral function of w only (Art. 30, iii.) ; let Pp (w)=a+bw+cw* + dw". 
F (0) F @-") 
F (@-™) 
fore PW (w)=W (w*); i.e. (b6—d) (1—1t) w+ 2ci=0, or c=0, D=d, and py (w)= 
a+5(1+%)o, p(w’) =a+b (1-2) w !; so that p= (w) x (o j= 
5 
a?4-902 (Art. 30, iv.). Again, J (")=a+8 (9"-+y™) = — pag mod p 
19n 
primitive root of the congruence #?— =1,mod p. Theny (w)= 
The function 
is not changed, if for @~” we write 6—*”; there- 
(Art. 30, v.), and p (y")=a—b (y"+y") = aii ee whence 
a=-—3 _ Hon To show that a= —1, mod 4, we observe that by the 
Mn . T4n 
definition of the function J, J (w)= Zw" *, y, and y, representing any two 
numbers of the series 1,2, ...  —2, which satisfy the congruence y’1+ y”= 1, 
mod p. Hence a= = (—1)”*"!, where y, is one of the numbers 1, 2,.. 2n—1, 
and n,, y, satisfy the congruence y+ y/?==1, mod p. Let @ be any one of 
the numbers 1, 2,..—1, and let A, B be the values of y’? corresponding to 
the values n—o, n+o of »,; then AX B=(1—y*%"”) x (1—y4 @F) = 
—y tf OF) x (1—y'@t), mod p; therefore A x B is a quadratic residue of 
p, and the values of y, corresponding to the values »—o, n+< of n, are either 
both even or else both uneven; also, if »,=n, y’? == 2, mod p, and y, is even, 
because 2\ 1. Let & be the number of values of n,, included in the series 
p 
1, 2, ..n—1, for which y,+7, isuneven; then a=3(—1)”*%=2(n—1)— 
Ale+(—1)”; i. e. a= —1, mod 4. 
We might also determine « in the equation p=«*+ 2y’ by the congruence 
@=(—1)"3 a mod p, or by the congruence w=2” x4 Pee 
mod p. ‘These determinations, which have been given by M. Stern, may 
EF (0-”) EF (Oe) 
F (Os 
a—n\72 
oR or may be deduced from the formula of Jacobi. The formuls for 
the determination of w in the first two theorems also admit of various modi- 
fications. It will be observed that, in the first, y is determined by a con- 
gruence as well as x. This determination is obtained by a comparison of the 
either be obtained directly by considering the functions 
2 
two congruences 1 +5 = 0, mod p,1+(I12n)’ = 0, mod p (the latter arising 
from Sir J. Wilson’s theorem); with regard to it Gauss observes, “ quum 
insuper noverimus quo signo affecta prodeat radix quadrati imparis, eo scilicet 
ut semper fiat forme 4m-+1, attentione perdignum est, quod simile criterium 
generale respectu radicis quadrati paris hactenus inveniri non potuerit. Quale 
si quis inveniat et nobiscum communicet magnam de nobis gratiam feret,” 
