In additive number theory, Fermat's theorem
on sums of two squares states that an odd
prime p can be expressed as:
p
=
x
2
+
y
2
,
{\displaystyle p=x^{2}+y^{2},}
with x and y integers, if and only if
p
≡
1
(
mod
4
)
.
{\displaystyle p\equiv 1{\pmod {4}}.}
The prime numbers for which this is true are
called Pythagorean primes.
For example, the primes 5, 13, 17, 29, 37
and 41 are all congruent 
to 1 modulo 4, and they can be expressed as
sums of two squares in the following ways:
5
=
1
2
+
2
2
,
13
=
2
2
+
3
2
,
17
=
1
2
+
4
2
,
29
=
2
2
+
5
2
,
37
=
1
2
+
6
2
,
41
=
4
2
+
5
2
.
{\displaystyle 5=1^{2}+2^{2},\quad 13=2^{2}+3^{2},\quad
17=1^{2}+4^{2},\quad 29=2^{2}+5^{2},\quad
37=1^{2}+6^{2},\quad 41=4^{2}+5^{2}.}
On the other hand, the primes 3, 7, 11, 19,
23 and 31 are all congruent to 3 modulo 4,
and none of them can be expressed as the sum
of two squares. This is the easier part of
the theorem, and follows immediately from
the observation that all squares are congruent
to 0 or 1 modulo 4.
Albert Girard was the first to make the observation,
describing all positive integral numbers (not
necessarily primes) expressible as the sum
of two squares of positive integers; this
was published in 1625. The statement that
every prime p of the form 4n+1 is the sum
of two squares is sometimes called Girard's
theorem. For his part, Fermat wrote an elaborate
version of the statement (in which he also
gave the number of possible expressions of
the powers of p as a sum of two squares) in
a letter to Marin Mersenne dated December
25, 1640: for this reason this version of
the theorem is sometimes called Fermat's Christmas
theorem.
Since the Diophantus identity implies that
the product of two integers each of which
can be written as the sum of two squares is
itself expressible as the sum of two squares,
by applying Fermat's theorem to the prime
factorization of any positive integer n, we
see that if all the prime factors of n congruent
to 3 modulo 4 occur to an even exponent, then
n is expressible as a sum of two squares.
The converse also holds. This equivalence
provides the characterization Girard guessed.
== Proofs of Fermat's theorem on sums of two
squares ==
Fermat usually did not write down proofs of
his claims, and he did not provide a proof
of this statement. The first proof was found
by Euler after much effort and is based on
infinite descent. He announced it in two letters
to Goldbach, on May 6, 1747 and on April 12,
1749; he published the detailed proof in two
articles (between 1752 and 1755). Lagrange
gave a proof in 1775 that was based on his
study of quadratic forms. This proof was simplified
by Gauss in his Disquisitiones Arithmeticae
(art. 182). Dedekind gave at least two proofs
based on the arithmetic of the Gaussian integers.
There is an elegant proof using Minkowski's
theorem about convex sets. Simplifying an
earlier short proof due to Heath-Brown (who
was inspired by Liouville's idea), Zagier
presented a one-sentence proof of Fermat's
assertion.
And more recently Christopher gave a partition-theoretic
proof.
== Related results ==
Fermat announced two related results fourteen
years later. In a letter to Blaise Pascal
dated September 25, 1654 he announced the
following two results for odd primes
p
{\displaystyle p}
:
p
=
x
2
+
2
y
2
⇔
p
≡
1
or
p
≡
3
(
mod
8
)
,
{\displaystyle p=x^{2}+2y^{2}\Leftrightarrow
p\equiv 1{\mbox{ or }}p\equiv 3{\pmod {8}},}
p
=
x
2
+
3
y
2
⇔
p
≡
1
(
mod
3
)
.
{\displaystyle p=x^{2}+3y^{2}\Leftrightarrow
p\equiv 1{\pmod {3}}.}
He also wrote:
If two primes which end in 3 or 7 and surpass
by 3 a multiple of 4 are multiplied, then
their product will be composed of a square
and the quintuple of another square.In other
words, if p, q are of the form 20k + 3 or
20k + 7, then pq = x2 + 5y2. Euler later extended
this to the conjecture that
p
=
x
2
+
5
y
2
⇔
p
≡
1
or
p
≡
9
(
mod
20
)
,
{\displaystyle p=x^{2}+5y^{2}\Leftrightarrow
p\equiv 1{\mbox{ or }}p\equiv 9{\pmod {20}},}
2
p
=
x
2
+
5
y
2
⇔
p
≡
3
or
p
≡
7
(
mod
20
)
.
{\displaystyle 2p=x^{2}+5y^{2}\Leftrightarrow
p\equiv 3{\mbox{ or }}p\equiv 7{\pmod {20}}.}
Both Fermat's assertion and Euler's conjecture
were established by Lagrange.
A generalization of Fermat's theorem, the
sum of two squares theorem, characterizes
the integers (not necessarily prime) that
can be expressed as the sum of two squares.
They are exactly the integers in which each
prime that is congruent to 3 mod 4 appears
with an even exponent in the prime factorization
of the number.
== See also ==
Proofs of Fermat's theorem on sums of two
squares
Legendre's three-square theorem
Lagrange's four-square theorem
Landau–Ramanujan constant
Thue's lemma
== Notes
