ON THE THEORY OF NUMBERS. 781 
by 2?+7y*. M. Stern found by induction that the value of w in the equation 
p=8n4+3=a"+4 27° 
satisfies the congruences 
1 W4n+1 
——— dp, «=(—1)", 14#*; 
D 2 fn. T3n+1 Ul ih Calvan 
and Eisenstein succeeded in demonstrating this theorem, as well as the two 
following t :— 
“Tf p=7n4+2=2°+4+7y’, 
1 [sn 
2 Tn. TL2n’ 
“Tf p= in+4=2°+4+7,’, 
me lon I 
SS = ’ 
2 Tin. W2n+1 
These demonstrations are obtained by expressing the prime number p as 
the product of two complex factors, composed of 8th or 7th roots of unity. 
But the decomposition of p is no longer supplied by the formula of Art. 30 ; 
nor are the complex factors included in the definition of the functions v, 
which have been considered in Art. 50 and in the last Article. 
If p=8n-+3 is a real prime, p is also a prime in the theory of complex num- 
bers of the form a+6i; let y be a primitive root of p in that theory, and let 
yJ=1+ iz, mod p, z representing one of the real integers, 0, 1,..y—1. Also 
let Y(w)= wy, w denoting a primitive 8th root of unity, and the summa- 
tion extending to every value of y. Eisenstein establishes the equations 
Wolo )=p, W(w)=v(w*); whence (w) is of the form A+B(1+7)u, 
and p=(w)y(w—!)=A?2+2B*. To find the residue of A, mod p, let 
e=}(p?—1)=3n+1+np; 
and write successively y* and +*¢ for w in the function {(w). We find 
Wy) =Dy9 = X(1 +722)? = (1 +2%)*"41(1—z)", mod p, 
because in general (a+67)?=(a—d?), mod p. In this expression no power 
of z has an exponent divisible by p—1; but 32_?"' 
different from zero, and is a multiple of p—1; therefore (7°) ==0, mod p. 
Again, because 5e=7n+2+(5n+1)p, Wy?) = 30 +2)"? (1—ziy" 4}, 
mod p; in this expression the coefficient C of ?~? is 
N7n+2 Hbn+1 
Tp W7n+2—p Op’ 15n+1—p” 
where p+’ =p—1, and the summation extends from p»=3n+1 to p=7n+42. 
Writing 38n+1-+-y for p, 5n4+1—y for p’, and observing that 
Ip. Mu=(—1y mod p, 
we Sie 
modp; #==3, mod 7; 
i 
mod p; #=2, mod 7.” 
*—=(, mod p, unless @ is 
a lal 
we find 
Gay O7n+2.05n4+1  N7n+2.05n+1  yadng41 TW4n41 
oy Pre ty. ly, Ti4ne  iee vee at Uy Na 
— Win+2.05n4+1 ogg W4n+1 d 
re T4n+1 x2 ~ Tn. 138n +1’ aereite 
* Crelle, vol. xxxii. p. 89. We enunciate the latter part of the theorem in the form in 
which it has been given by Hisenstein. 
+ Crelle, vol. xxxvii. p. 97. 
