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In this paper Don Zagier constructs a really short proof of the ver...
Don Zagier, born in 1951, is an American mathematician whose main a...
First we should recall what is an involution. An involution is a fu...
Zagier starts by defining two involutions $$f(x,y,z) = (x,z,y)$$...
To calculate the fixed points of $g$ we'll have to solve the equati...
Lemma: The cardinalities of a finite set and of its fixed-point se...
To sum up: Zagier starts by creating a function $g$ which is an ...
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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