THE

TEACHING OF

MATHEMATICS

EDITED BY MELVIN

HENRIKSEN AND

STAN WAGON

A

One-Sentence Proof

That Every Prime

p

1

(mod 4)

Is a Sum of Two

Squares

D.

ZAGIER

Departmenit

of Mathematics,

University of

Maryland, College Park,

MD 20742

The

involution on

the finite set

S

=

{(x,y,z)

E

rkJ3:

X2

+ 4yz

=

p

}

defined by

((x

+

2z,

z, y-x-z)

if

x

<y-z

(x,y,z)

|->4 (2y

-

x,

y, x

-

y

+

z)

if

y

- z <

x < 2y

I(x

-

2y, x

-y

+

z, y)

if

x

>

2y

has

exactly one fixed

point, so

ISI

is odd and the

involution

defined by

(x,y,z)

-

(x,z,y)

also has a fixed

point. O

This

proof is a

simplification of

one due to

Heath-Brown [1]

(inspired,

in

turn, by

a

proof

given by

Liouville).

The verifications of the

implicitly

made assertions-that

S

is

finite and

that the

map

is

well-defined and

involutory

(i.e., equal

to its own

inverse)

and has

exactly

one

fixed

point-are

immediate and

have been left to the

reader.

Only the last

requires that p

be a prime of

the form 4k +

1, the fixed

point

then

being (1,1,k).

Note

that the proof

is not

constructive: it does

not give a

method to

actually find

the

representation

of

p

as a

sum

of two

squares.

A

similar

phenomenon

occurs with

results

in

topology and

analysis that

are proved

using

fixed-point theorems.

Indeed,

the basic

principle we used:

"The

cardinalities of a finite set and of its

fixed-point

set under

any

involution have the same

parity,"

is

a combinatorial

analogue

and

special

case of

the

corresponding

topological

result:

"The Euler characteristics

of

a

topological

space

and of

its

fixed-point

set under

any

continuous involution have

the same

parity."

For a

discussion

of

constructive

proofs

of the

two-squares

theorem,

see the

Editor's Corner elsewhere

in

this issue.

REFERENCE

1. D. R.

Heath-Brown, Fermat's

two-squares

theorem, Invariant

(1984) 3-5.

Inverse

Functions

and their

Derivatives

ERNST SNAPPER

Department of

Mathematics

and Computer

Science,

Dartmouth College,

Hanover, NH 03755

If the concept

of inverse

function

is introduced

correctly,

the usual rule

for its

derivative

is visually

so obvious, it barely

needs

a

proof.

The reason

why

the

standard,

somewhat

tedious

proofs are

given is

that

the inverse

of a function

f(x)

is

144

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