Number Theory • Topic 27

Sums of Two Squares

Which integers can be written as the sum of two squares ( $n = a^{2} + b^{2}$ )? This classic problem links arithmetic to the geometry of the Gaussian integers.

Fermat's Theorem on Sums of Two Squares

Theorem (Fermat's Theorem)

An odd prime

p

can be written as

p = a^{2} + b^{2}

if and only if:

p \equiv 1 (mod 4)

Primes $p \equiv 3 (mod 4)$ (like 3, 7, 11) cannot be written as sums of two squares.
The prime 2 can be written as $1^{2} + 1^{2}$ .

General Integers

Using the Brahmagupta–Fibonacci identity:

(a^{2} + b^{2}) (c^{2} + d^{2}) = (a c - b d)^{2} + (a d + b c)^{2}

The product of sums of two squares is a sum of two squares.

Theorem (Sum of Two Squares Condition)

A positive integer

n

can be written as a sum of two squares if and only if in the prime factorization of

n

, every prime of the form

4 k + 3

occurs to an even power.

Example

n = 45 = 3^{2} \cdot 5

Prime 3 ( $4 k + 3$ form) has exponent 2 (even).
Prime 5 ( $4 k + 1$ form) has exponent 1.
Yes, 45 is a sum of two squares ( $3^{2} + 6^{2}$ ).

Practice Problems

Exercise (Problem 1)

Express

p = 13

and

p = 29

as sums of two squares.

Exercise (Problem 2)

Determine if

n = 245

can be written as a sum of two squares.

Exercise (Problem 3)

Prove that no integer of the form

4 k + 3

can be a sum of two squares.