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A One-Sentence Proof That Every Prime $p\equiv 1(\mod 4)$ Is a Sum of Two SquaresAuthor(s): D. ZagierSource: The American Mathematical Monthly, Vol. 97, No. 2 (Feb., 1990), p. 144Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2323918.

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THE

TEACHING OF

MATHEMATICS

EDITED BY MELVIN

HENRIKSEN ND TANWAGON

A One-Sentence roofThat Every rimep 1 (mod4)

Is a Sum of Two

Squares

D.

ZAGIER

Departmenit

fMathematics,

niversityfMaryland, ollegePark,

MD 20742

The involution n

the finite et S

=

{(x,y,z) E

rkJ3:

2

+

4yz

=

p defined y

((x

+

2z,

z, y-x-z)

if

x

4

(2y

-

x,

y, x

-

y

+

z) if

y

- z 2y

has exactlyone fixed

point, o

ISI

is odd and the nvolution

efined y

(x,y,z)

-

(x,z,y)

also has a fixed

oint.O

This proof s a

simplificationf

one due toHeath-Brown1]

inspired,

n

turn, y

a

proof

givenby

Liouville).

The verificationsf the

mplicitly

ade assertions-that

S

is

finite nd

that the

map

is

well-definednd

involutoryi.e., equal

to its own

inverse) nd has

exactly

ne

fixedpoint-are

immediate nd

have been left o the

reader.

Only the ast requires hat p

be a prime f

the form k + 1, the fixed

point

then

being 1,1,k).

Note thattheproof

s not constructive:t does

not givea

method o actually ind

the

representation

f

p as a

sum

of two

squares.

A

similar henomenon ccurswith

results

n

topology nd analysis hat

re proved

usingfixed-pointheorems.

ndeed,

the basic

principlewe used:

The cardinalities f a finite et and of its

fixed-point

set under

any

involution ave the same

parity,

s

a combinatorial

nalogue

and

special

case of

the

corresponding

opological

esult:

The Euler characteristics

f

a

topologicalspace

and of

its

fixed-point

et under

any

continuous nvolution ave

the same

parity.

For a discussionof constructive roofsof the two-squares heorem, ee the

Editor's Cornerelsewhere

n

this ssue.

REFERENCE

1. D. R.

Heath-Brown, ermat's wo-squares

heorem, nvariant

1984) 3-5.

Inverse

unctions

nd their erivatives

ERNST SNAPPER

Department fMathematics

nd Computer

cience,Dartmouth ollege,

Hanover,NH 03755

If the concept of inverse

unction

s introduced orrectly,

he usual rule

for ts

derivative

s visually

so obvious, it barelyneeds

a

proof.

The reason

why

the

standard, omewhat

ediousproofs re

given s

that he nverse

f a function

(x)

is

144