Introduction

• Many numbers can be written as a sum of squares, such as
  • $5 = 1^2 + 2^2$
  • $25 = 3^2 + 4^2$
• Others cannot, such as $3$ or $7$.
• This article explores:
  • Which numbers can be written as a sum of two squares
  • Why some numbers cannot
  • Why every number can be written as a sum of four squares
• Along the way, we’ll see patterns, geometry, and a bit of number theory.

What Does It Mean to Be a Sum of Squares?

• A number $n$ is a sum of two squares if $$n = a^2 + b^2$$ for some whole numbers $a$ and $b$.
• A number is a sum of four squares if $$n = a^2 + b^2 + c^2 + d^2.$$
• We usually allow $0^2$ as a square.
• Examples:
  • $1 = 1^2 + 0^2$
  • $2 = 1^2 + 1^2$
  • $50 = 1^2 + 7^2$

Geometric Intuition: Distances and the Plane

• The expression $a^2 + b^2$ appears naturally in geometry:
  • It is the square of the distance from $(0,0)$ to $(a,b)$.
• Thinking geometrically:
  • A number is a sum of two squares if it is the squared distance to some lattice point.
• Visual intuition:
  • Points with integer coordinates form a grid.
  • We are asking: Which distances from the origin land exactly on a grid point?

Exploring Small Examples

Try listing numbers and checking whether they are sums of two squares:

• $1 = 1^2 + 0^2$
• $2 = 1^2 + 1^2$
• $3$ — no combination of squares works
• $4 = 2^2 + 0^2$
• $5 = 1^2 + 2^2$
• $6$ — no combination works
• $7$ — no combination works
• $8 = 2^2 + 2^2$
• $9 = 3^2 + 0^2$
• $10 = 1^2 + 3^2$

Patterns begin to appear, but they’re subtle.

Patterns Begin to Emerge

Observations from small numbers:

• Many numbers of the form $4k+1$ work (e.g., $5, 13, 17$).
• Many numbers of the form $4k+3$ do not work (e.g., $3, 7, 11$).
• But there are exceptions, so we need a deeper rule.
• Multiplying sums of squares often gives another sum of squares:
  • If $m = a^2 + b^2$ and $n = c^2 + d^2$, then $$mn = (a^2 + b^2)(c^2 + d^2).$$
  • Which is equivalent to: $$mn = (ac - bd)^2 + (ad + bc)^2.$$

This multiplication property is a major clue.

Prime Numbers and Sums of Two Squares

Prime numbers behave in a very structured way:

• Some primes are themselves sums of two squares:
  • $5 = 1^2 + 2^2$
  • $13 = 2^2 + 3^2$
• Others are not:
  • $3, 7, 11, 19$ cannot be written as $a^2 + b^2$.

The key pattern:

• A prime $p$ is a sum of two squares exactly when $$p \equiv 1 \pmod{4}$$ or $$p = 2$$

Fermat’s Two‑Square Theorem

Fermat’s theorem states:

• A prime $p$ can be written as $a^2 + b^2$
  if and only if $$p = 2 \quad \text{or} \quad p \equiv 1 \pmod{4}.$$

Key ideas (without proof):

• Primes of the form $4k+3$ have a property that prevents them from splitting into two squares.
• Primes of the form $4k+1$ behave differently and allow such representations.

Composite Numbers: Building Up from Primes

Once we know how primes behave, we can understand all numbers.

A composite number $n$ is a sum of two squares if and only if:

• In the prime factorization of $n$, every prime of the form $4k+3$ appears with an even exponent.

Example:

• $45 = 3^2 \cdot 5$
  • The prime $3$ (which is $4k+3$) appears with exponent $2$ (even)
  • So $45$ is a sum of two squares: $$45 = 3^2 + 6^2.$$

Why Some Numbers Fail to Be Sums of Two Squares

Numbers fail for a simple reason:

• If a prime $p \equiv 3 \pmod{4}$ appears with an odd exponent in $n$, then $n$ cannot be a sum of two squares.

Examples:

• $3$ fails because $3^1$ has odd exponent.
• $21 = 3 \cdot 7$ fails because both primes are $4k+3$ and appear with odd exponents.
• $6 = 2 \cdot 3$ fails because $3$ appears once.

Four Squares: A Surprising Universal Property

Unlike the two‑square case, the four‑square case is dramatically simpler:

• Every whole number can be written as a sum of four squares.

Examples:

• $7 = 4 + 1 + 1 + 1$
• $23 = 16 + 4 + 1 + 1$
• $0 = 0 + 0 + 0 + 0$

This universality is one of the most beautiful facts in number theory.

Lagrange’s Four‑Square Theorem

Lagrange proved in 1770:

• Every natural number $n$ can be written as $$n = a^2 + b^2 + c^2 + d^2.$$

Key ideas (informal):

• Squares grow quickly, so only a few are needed.
• Clever algebraic identities show that sums of four squares multiply nicely.
• This allows building representations for all numbers.

Visualizing Sums of Four Squares

Ways to picture four squares:

• Think of a point in 4‑dimensional space: $(a,b,c,d)$.
• The number $n$ is the squared distance from the origin: $$n = a^2 + b^2 + c^2 + d^2.$$
• Every “radius” in 4D space hits some lattice point.

This geometric viewpoint explains why four squares always suffice.

Connections to Modular Arithmetic

Modular arithmetic helps explain patterns:

• Squares modulo $4$ can only be $0$ or $1$.
• This explains why primes of the form $4k+3$ behave differently.
• Modulo $8$ also gives insight into sums of three squares (a related topic).

Historical Notes and Mathematical Personalities

• Fermat: first stated the two‑square theorem.
• Euler: provided the first proofs and identities.
• Lagrange: proved the four‑square theorem.
• Gauss: developed the theory of quadratic forms, generalizing these ideas.

These results were stepping stones toward modern number theory.

Common Misconceptions and Pitfalls

• “If $n$ is a sum of two squares, then so is $n+1$.”
  • False: $4$ works, $5$ works, but $6$ does not.
• “Odd numbers cannot be sums of two squares.”
  • False: $5, 13, 17$ all work.
• “Only primes matter.”
  • Composite numbers require careful factorization.

Exercises

Try these to reinforce the ideas:

1. Determine whether $21$ can be written as a sum of two squares. Explain why or why not.

   Solution

   Problem: Determine whether $21$ can be written as a sum of two squares.

   • Factorization: $$21 = 3 \cdot 7.$$
   • Both $3$ and $7$ are primes of the form $4k+3$, each with exponent $1$ (odd).
   • Rule: If any prime $\equiv 3 \pmod{4}$ appears with an odd exponent, the number is not a sum of two squares.

   Answer: $21$ cannot be written as a sum of two squares.

2. Find all representations of $25$ as a sum of two squares.

   Solution

   Problem: Find all representations of $25$ as a sum of two squares.

   We look for integer solutions to $$a^2 + b^2 = 25.$$ Check small squares:

   • $0^2 + 5^2 = 25$
   • $3^2 + 4^2 = 9 + 16 = 25$

   Including sign changes and order:

   • $(a,b) = (\pm 5, 0), (0, \pm 5)$
   • $(a,b) = (\pm 3, \pm 4), (\pm 4, \pm 3)$

   Answer: Up to order and sign, the representations are $$25 = 5^2 + 0^2 = 3^2 + 4^2.$$

3. Show that $45$ is a sum of two squares by finding explicit $a$ and $b$.

   Solution

   Problem: Show that $45$ is a sum of two squares by finding explicit $a$ and $b$.

   • Factorization: $$45 = 3^2 \cdot 5.$$
   • The prime $3 \equiv 3 \pmod{4}$ appears with exponent $2$ (even), so $45$ can be a sum of two squares.

   Try small squares:

   • $6^2 = 36$, and $45 - 36 = 9 = 3^2$
     So $$45 = 6^2 + 3^2.$$

   Answer: One representation is $$45 = 6^2 + 3^2.$$

4. Factor $117$ and decide whether it is a sum of two squares.

   Solution

   Problem: Factor $117$ and decide whether it is a sum of two squares.

   • Factorization: $$117 = 9 \cdot 13 = 3^2 \cdot 13.$$
   • Here, $3 \equiv 3 \pmod{4}$ appears with exponent $2$ (even).
   • $13$ is a prime with $13 \equiv 1 \pmod{4}$, which is allowed.
   • Therefore, $117$ can be written as a sum of two squares.

   We can even find one representation:

   • $117 = 9 \cdot 13$
     and $13 = 2^2 + 3^2$, $9 = 3^2 + 0^2$.
   • Using the product formula: $$(3^2 + 0^2)(2^2 + 3^2) = (3\cdot 2 - 0\cdot 3)^2 + (3\cdot 3 + 0\cdot 2)^2 = 6^2 + 9^2 = 36 + 81 = 117.$$

   Answer: Yes, $117$ is a sum of two squares: $$117 = 6^2 + 9^2.$$

5. Write $23$ as a sum of four squares.

   Solution

   Problem: Write $23$ as a sum of four squares.

   Try to decompose $23$:

   • $4^2 = 16$, remainder $7$.
   • $7$ can be written as $4 + 1 + 1 + 1 = 2^2 + 1^2 + 1^2 + 1^2$.

   So: $$23 = 16 + 4 + 1 + 1 = 4^2 + 2^2 + 1^2 + 1^2.$$ Answer: $$23 = 4^2 + 2^2 + 1^2 + 1^2.$$

6. Prove that no number of the form $4k+3$ can be written as $a^2 + b^2$ when $k$ is small (try $k=0,1,2$).

   Solution

   Problem: Prove that no number of the form $4k+3$ can be written as $a^2 + b^2$ when $k$ is small (try $k=0,1,2$).

   We check $n = 4k+3$ for $k = 0,1,2$:

   • For $k=0$: $n = 3$
     • Squares modulo $4$ are $0$ or $1$.
     • Possible sums: $0+0=0$, $0+1=1$, $1+1=2$ (mod $4$).
     • None give $3$ (mod $4$), so $3$ is not a sum of two squares.
   • For $k=1$: $n = 7$
     • Same reasoning: any $a^2 + b^2$ is $0,1,$ or $2$ modulo $4$, never $3$.
     • So $7$ is not a sum of two squares.
   • For $k=2$: $n = 11$
     • Again, $a^2 + b^2$ cannot be $3$ modulo $4$, so $11$ is not a sum of two squares.

   Answer: For $3, 7, 11$ (all $4k+3$), no representation as $a^2 + b^2$ exists, consistent with the general rule.

7. Find a number that is a sum of four squares but not a sum of two squares.

   Solution

   Problem: Find a number that is a sum of four squares but not a sum of two squares.

   We know:

   • Every number is a sum of four squares.
   • Numbers with a prime $\equiv 3 \pmod{4}$ to an odd power are not sums of two squares.

   Take $7$:

   • $7$ is not a sum of two squares (from Exercise 6).
   • But $$7 = 4 + 1 + 1 + 1 = 2^2 + 1^2 + 1^2 + 1^2.$$

   Answer: $7$ is one example: $$7 = 2^2 + 1^2 + 1^2 + 1^2,$$ but it is not a sum of two squares.

8. Explore: How many ways can $50$ be written as a sum of two squares?

   Solution

   Problem: How many ways can $50$ be written as a sum of two squares?

   We solve $$a^2 + b^2 = 50.$$ Check small squares:

   • $1^2 + 7^2 = 1 + 49 = 50$
   • $5^2 + 5^2 = 25 + 25 = 50$

   So, up to order and sign, we have:

   • $50 = 1^2 + 7^2$
   • $50 = 5^2 + 5^2$

   Including signs and swapping $a$ and $b$:

   • $(\pm 1, \pm 7)$, $(\pm 7, \pm 1)$
   • $(\pm 5, \pm 5)$

   If we count distinct unordered pairs $\{a,b\}$ with $a,b \ge 0$, there are two:

   • $\{1,7\}$ and $\{5,5\}$.

   Answer: There are two essentially different representations: $$50 = 1^2 + 7^2 = 5^2 + 5^2.$$