SECTION III., 1900 za] Trans. R. S. C. 
VIII.—The Number of Representations of a Number as the Sum of Two 
Squares. 
By J. H. McDonatp, B.A., University of Toronto. 
(Communicated by Prof. Baker and read May 29, 1900.) 
Let m be an odd positive number, not a square. Then it has been 
shown (Dirichlet, Zahlentheorie, p. 229) that the number of representa- 
d— 1 
2 

tions of m by the form 2° + 7° of determinant —1 is 42 (—1) 
d a divisor of m or 4 (M—N) where M and Ware the number of divisors 
of m of the forms 4 À + 1, 4 À + 3 respectively. 
(N. B—If x, y be a representation of m, so also are + x + y; 
+ y +2, so that 8 representations give only one decomposition.) 
Dirichlet deduces this result as a corollary from general results, but 
it may be proved directly in such a way as to make it more intuitive. 
We have to enumerate the improper as well as the proper represen- 
tations of m. 
Let 2? + y? = m, and x = dz’, y = dy’, then d* must divide m; let 
m= dm. Then x’? + y’* = m'is a proper representation of m if d is 
the greatest common divisor of x and y, which we shall suppose. 
The theory of representation gives us the following results : 
The number m’ is not representable by x? + y’ if it contains any 
prime factor of the form 4A + 3: for —1 must be a quadratic residue of 
every prime factor of m’. 
If y denotes the number of distinct prime factors of m', supposed all 
of the form 4A + 1, then the congruence z* + 1 = 0 (mod m’) admits 27 
roots, and the number m’ has 4-2” representations, which are proper by 
the form x’ + y. Dirichlet, Zahlentheorie, pp. 87, 152, 165. 
We have then as the whole number of representations, proper and 
improper, of m by a? + y’, the sum 4 2". Where v is the number of 
ad’: m 
distinct primes dividing e and d’?e = m; d? being a square divisor of m 
which contains all the primes of the form 4 À + 3. 
We may write this sum as 42 2” — 4> 2” the first summation re- 
De dom 
ferring to all the square divisors of m, the second to those which do not 
contain all the primes of the form 4 A + 3. 
We note first that > 2” = the number of divisors of m. For let D° 
a: m 
be a square divisor of m and let m = D’e, Then each of the 2” num- 
