The following constitutes the essential text of a complete research article; I have
omitted only some comments at the end concerning the history of this type of argument.
The author reproves a famous result. He builds his proof into a single sentence
as simply a tour-de-force. In fact, he has left many straightforward steps for
the reader to verify.
1. As an exercise in critical reading, list all the implicit claims that the
reader must verify in order to accept this argument as a proof.
2. As an exercise in logic and algebra, supply all the details necessary to
support these claims. Package all this as a long-winded rewrite of Zagier's
article written so that any high school algebra student could easily read it
with comprehension.
You should expect to expand Zagier's single sentence to a full page or more.
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A One-Sentence Proof That Every Prime p congruent to 1 modulo 4 Is a Sum of Two Squares
D. Zagier
Department of Mathematics, University of Maryland, College Park, MD 20742
The involution on a finite set S = {(x,y,z) \in N^3 : x^2 +4yz = p } defined by
( x+2z, z, y-x-z ) if x < y-z
(x,y,z) ---> { ( 2y-x, y, x-y+z ) if y-z < x < 2y
( x-2y, x-y+z, y ) if x > 2y
has exactly one fixed point, so |S| is odd and the involution defined by
(x,y,z) ---> (x,z,y)
also has a fixed point.
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Glossary:
Cardinality: We write |S| for the number of elements the set S contains.
This has a clear meaning for a finite set S.
Congruence: We say we have integers a and b congruent modulo n if
n divides a - b. We often abbreviate "modulo" as "mod".
We can also say that a and b leave the same remainder when
divided by n (provided we take care to arrange for a non-negative
remainder less than n).
Fixed point: Given a function f:A ---> A from any set A to itself, we call
x \in A a fixed point if f(x) = x. In this case we sometimes
say "f does not move x" or "f fixes x."
Involution: Any function from any set to that same set which equals its inverse.
For an involution f:A--->A, f(f(x)) = x for all x \in A and conversely
this characterizes f as an involution.