2. - American Mathematical Society

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 101. Number 3, November 1987

A SIMPLE PROOF OF JACOBI'S FOUR-SQUARETHEOREM M. D. HIRSCHHORN (Communicated

by Larry J. Goldstein)

Abstract. Jacobi's four-square theorem, which gives the number of representations of a positive integer as a sum of four squares, is shown to follow simply from the triple-product identity.

1. Recently [2], I showed how one can obtain Jacobi's two-square theorem from the triple-product identity. In this note I show that the triple-product identity also gives Jacobi's four-square theorem: Theorem 1. The number rA(n) of representations of four squares is given by

r4(«) = 8

£

of the positive integer n as a sum

d.

l

x4^^-

n (i + ^2"-i)2(i -x2")) dx „Vf yj

Employing the product rule to evaluate the derivatives, we find

non» 1

x"t = n (i + *2"-')2(i - *2")(i + *2")2(i - x2") n» 1

L

. ~ —

\

2«jc2" i

i

2«

n » 1 x + ■*

. v "

2«x2" i _

2n

w» 1 1

■*

,

- n o + *2")2(i - *a")o + *2"-i)2(i - x2") n>l

x|>£ (1> 1

("-')"';-'-4£ 1 _|_

1 + A

2n-l

w —

1 _

»> 1 1

x

2((

or,

noí¡> i

*")6 = n o + x2"_i)2(i + *2")2(i - x2")2 «>i x(l-8El(2W"1)x2"" 1 + x2""1

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^ 1 + x2"

438

M. D. HIRSCHHORN

Dividing both sides by

n o + x")\i - x"f = n o + *")2(i- x2")2

n Ss 1

n> 1

= n

a + x2""1)2^

+ *2")2(i -x2")2,

n»l

we obtain

I1"*")4-1

8£^(2w"1^2'"'

„Vi U + x"}

X\ \

2nx2r

1 + *2"-1

1 + x2"

Now, it is a simple consequence of the triple-product identity that

n (££) -£(-i)v. so we have

- 00

/

1 + x2""1

(I » 1

1 + x2n

Putting -x for x, we obtain

£,-),.i+«s( 1 1I

1 - X2"

1 + X2"

4nx4"

n>l *■~ x

or.

nx

£*"* =i + 8£r i from which Theorem

1 follows directly.

References 1. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon

Press,

Oxford. 2. Michael

D. Hirschhorn,

A simple proof of Jacobi's

two-square theorem. Amer. Math. Monthly 92

(1985), 579-580. School of Mathematics, University South Wales, Australia 2033

of New South Wales, P. O. Box 1, Kensington,

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