PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 4 JULY 15, 1918 Number 7 
ON THE REPRESENTATION OF A NUMBER AS THE SUM OF ANY 
NUMBER OF SQUARES, AND IN PARTICULAR OF 
FIVE OR SEVEN 
By G. H. Hardy 
Trinity College, Cambridge, England 
Communicated by E. H. Moore, May 21, 1918 
1. The formulae concerning the representation of a number as the sum of 
5 or 7 squares belong to one of the most unfamiliar and difficult chapters in 
the Theory of Numbers, and only one proof of them has been given. The 
proof depends on the general arithmetx theory of quadratic forms, initiated 
by Eisenstein and perfected by Smith and Minkowski. This theory, of 
which a systematic account will be found in the fourth volume of Bachmann's 
Zahlentheorie gives a complete solution of the problem of any number s of 
squares not exceeding 8. Beyond s = 8 it fails. 
When s is even there is an alternative method. This method, which de- 
pends on the theory of the elliptic modular functions, is much simpler in idea 
than the method of Smith and Minkowski; and it has another very important 
merit, that it can be used — within the limits of human capacity for calcula- 
tion — for any even value of s. Thus Jacobi solved the problem for 2, 4, 6 and 
8. In these cases the number of representations can be expressed in terms of 
the divisors of n. Suppose, e.g., that 5 = 8; and let us write, generally, 
oo 
l 
where q = e™ Then 
\l+q 1-q 2 1+q* 1 - q* ) 
and r 8 (n) is 16 2 5 3 if n is odd and 8 2 <5q — 8 2 ? if n is even, 8 denoting 
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