A One-Sentence Proof That Every Prime $p\equiv 1(\mod 4)$ Is a Sum of Two Squares
Author(s): D. Zagier
Source: The American Mathematical Monthly, Vol. 97, No. 2 (Feb., 1990), p. 144
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2323918 .
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THE TEACHING OF MATHEMATICS
EDITEDBY MELVINHENRIKSEN
ANDSTANWAGON
A One-SentenceProofThat EveryPrimep
Is a Sum of Two Squares
1 (mod4)
D. ZAGIER
ofMathematics,
Departmenit
University
ofMaryland,CollegePark,MD 20742
The involutionon thefiniteset S = {(x,y,z) E
(x,y,z)
|->4
((x + 2z, z, y-x-z)
(2y - x, y, x - y + z)
I(x - 2y, x -y + z, y)
rkJ3:
X2 +
4yz = p } definedby
if x <y-z
if y - z < x < 2y
if x > 2y
has exactlyone fixedpoint,so ISI is odd and theinvolutiondefinedby (x,y,z) (x,z,y) also has a fixedpoint.O
This proofis a simplification
of one due to Heath-Brown[1] (inspired,in turn,by
a proofgivenby Liouville).The verifications
of theimplicitly
made assertions-that
S is finiteand that the map is well-definedand involutory(i.e., equal to its own
inverse)and has exactlyone fixedpoint-are immediateand have been leftto the
reader.Only thelast requiresthatp be a primeof theform4k + 1, thefixedpoint
thenbeing (1,1,k).
Note thattheproofis notconstructive:
it does not givea methodto actuallyfind
the representation
of p as a sum of two squares.A similarphenomenonoccurswith
resultsin topologyand analysisthatare provedusingfixed-point
theorems.Indeed,
the basic principlewe used: "The cardinalitiesof a finiteset and of its fixed-point
set under any involutionhave the same parity,"is a combinatorialanalogue and
of a
special case of the corresponding
topologicalresult:"The Euler characteristics
topological space and of its fixed-pointset under any continuousinvolutionhave
the same parity."
For a discussion of constructiveproofs of the two-squarestheorem,see the
Editor's Cornerelsewherein thisissue.
REFERENCE
1. D. R. Heath-Brown,Fermat'stwo-squarestheorem,Invariant(1984) 3-5.
InverseFunctionsand theirDerivatives
ERNST SNAPPER
College,Hanover,NH 03755
Science,Dartmouth
and Computer
DepartmentofMathematics
the usual rule forits
If the concept of inversefunctionis introducedcorrectly,
derivativeis visually so obvious, it barely needs a proof. The reason why the
standard,somewhattediousproofsare givenis thattheinverseof a functionf(x) is
144