Number Theory: An Introduction to Mathematics

XIII Connections with Number Theory..............................

1 SumsofSquares

In Proposition II.40 we proved Lagrange’s theorem that every positive integer can
be represented as a sum of 4 squares. Jacobi (1829), at the end of hisFundamenta
Nova, gave a completely different proof of this theorem with the aid of theta functions.
Moreover, his proof provided a formula for the number of different representations.
Hurwitz (1896), by developing further the arithmetic of quaternions which was used
in Chapter II, also derived this formula. Here we give Jacobi’s argument preference
since, although it is less elementary, it is more powerful.

Proposition 1The number of representations of a positive integer m as a sum of
4 squares of integers is equal to 8 times the sum of those positive divisors of m which
are not divisible by 4.

Proof From the series expansion

θ 00 ( 0 )=

∑

n∈Z

qn

2

we obtain

θ 004 ( 0 )=

∑

n 1 ,...,n 4 ∈Z

qn

2

1 +···+n

2

(^4) = 1 +

∑

m≥ 1

r 4 (m)qm,

wherer 4 (m)is the number of solutions in integersn 1 ,...,n 4 of the equation

n^21 +···+n^24 =m.

We will prove the result by comparing this with another expression forθ 004 ( 0 ).
We can write equation (43) of Chapter XII in the form

θ 10 ′(v)/θ 10 (v)−θ 01 ′(v)/θ 01 (v)=πiθ 002 ( 0 )θ 00 (v)θ 11 (v)/θ 01 (v)θ 10 (v).

Differentiating with respect tovand then puttingv=0, we obtain

θ 10 ′′( 0 )/θ 10 ( 0 )−θ 01 ′′( 0 )/θ 01 ( 0 )=πiθ 003 ( 0 )θ 11 ′( 0 )/θ 01 ( 0 )θ 10 ( 0 )=−π^2 θ 004 ( 0 ),

W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext,
DOI: 10.1007/978-0-387-89486-7_13, © Springer Science + Business Media, LLC 2009

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