~ 
Ni 
~t 
ON THE THEORY OF NUMBERS. 
or, since (20""— =")? = —A*, 
Apo) = (A+B)? +A (A—BY, 
W, (A) representing the number of numbers less than A and prime to A. We 
have next to determine the highest power of p dividing A+B and A—B, or, 
which is the same thing, A and B. By the principles indicated above, we 
have 
Ss 
A>6% 4+BE0"=p* ®, (6) 
wv, (@) 
xb 
Ase™ 4 Bye p* 2-1) 
¥_,() 
Writing in these equations y for @, and observing that the determinant 
(Sy) — = (y)?t, as wellas the four numbers ®, (y),b_,(y),¥,(7),¥_1(), 
is prime to p, we infer that the exponent of the highest power of p dividing 
A and B is the less of the two numbers = = Of these the former is the 
> 
a 
lesst; if therefore we write « and y for (—1)‘ (A+B) >. 
Xa 2b— Sa 
/ (A—B) p> respectively, our equation becomes 4p * =a?+Ay?. Also, 
» and 
* See Art. 96, ix. of this Report, or the note on Art. 104. 
+ Since D094 D5n—(—1)A, we have By"+4Sy?"= (—1)A, mod p; and since 
(26% — S6bn)?= —A, we have (Sy —Sy'”)?= —A, mod p. Thus the two factors of the 
determinant are each of them prime to p. 
The principle that any rational equation containing only powers of @ and integral num- 
bers may be changed into a congruence for the modulus p, if y be written in it for 0, has 
already been employed in this article. Its truth is evident, if we observe that the irre- 
ducible equation satisfied by 0, if considered as a congruence for the modulus , is satisfied 
by y. This principle is of more general application than a similar one which has been 
already employed in Art. 51 of this Report ; but its proof supposes the irreducibility of the 
equation of the primitive roots, which is not necessary to the proof of the principle of 
Art. 51. 
+ =b— a is certainly positive, because z 
” is equal to the number of improperly 
primitive classes of the negative determinant —A. Or (as it is desirable to avoid making 
as 
AWA 
use of this result here) =5— a is positive because (26 — =a) is the sum of the series 
ao 
2/2 a the summation extending to every value of prime to A, and the terms bein 
All a 2 & 
a 
taken in their natural order. This series is positive, because the series - (=) x4 eof 
which it is the limit, when p is diminished without limit, is certainly positive, being the reci- 
procal of the product n[ 1- (4) - a et in which the sign of multiplication extends to 
every prime qg not dividing A, and in which every factor is positive. The series 
: (x) 1 is one of those summed by Dirichlet in the memoir “ Recherches sur diverses 
n 
applications, &c.” (Crelle, vol. xxi. p. 141 ef seq.) : for the case in which A is a prime, he 
had already summed it in the memoir on the Arithmetical Progression (Memoirs of the 
Academy of Berlin for 1837, p. 55). Cauchy (Mémoires de l Académie des Sciences, vol. 
xvii. p. 673 et seq.) inverts Dirichlet’s process, ard transforms sums of the form = f(a) — 
=f (5) into infinite series. The transformation is effected by substituting for f(x), in the 
. 
