Every number can be written as the sum of four squares.  Prove it

The statement that every natural number can be written as the sum of four squares is known as Lagrange's Four Square Theorem. Here's a detailed proof of this theorem:

### Lagrange's Four Square Theorem
**Theorem:** Every natural number \( n \) can be written as the sum of four integer squares, i.e., for any \( n \in \mathbb{N} \), there exist integers \( a, b, c, \) and \( d \) such that:
\[ n = a^2 + b^2 + c^2 + d^2. \]

### Proof
The proof involves several steps and uses properties of modular arithmetic, the theory of quadratic forms, and properties of numbers.

#### Step 1: Base Cases
First, we verify the theorem for small values of \( n \).

- For \( n = 1 \), \( 1 = 1^2 + 0^2 + 0^2 + 0^2 \).
- For \( n = 2 \), \( 2 = 1^2 + 1^2 + 0^2 + 0^2 \).
- For \( n = 3 \), \( 3 = 1^2 + 1^2 + 1^2 + 0^2 \).
- For \( n = 4 \), \( 4 = 2^2 + 0^2 + 0^2 + 0^2 \).

#### Step 2: Induction Hypothesis
Assume that every number \( k \) where \( k < n \) can be expressed as the sum of four squares.

#### Step 3: Properties of Sums of Squares
We use the following identity for sums of four squares (a result from the theory of quadratic forms):
\[ (a^2 + b^2 + c^2 + d^2)(e^2 + f^2 + g^2 + h^2) = (ae - bf - cg - dh)^2 + (af + be + ch - dg)^2 + (ag - bh + ce + df)^2 + (ah + bg - cf + de)^2. \]

This identity shows that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

#### Step 4: Application of Modular Arithmetic
To complete the proof, we use modular arithmetic and the properties of sums of squares modulo 8.

- Any integer is congruent to one of \( 0, 1, 2, 3, 4, 5, 6, \) or \( 7 \mod 8 \).

We observe that any square number modulo 8 is congruent to 0, 1, or 4. Thus, the sum of four squares modulo 8 can be any integer between 0 and 7.

1. **\( n \equiv 0 \mod 8 \):**
   \[ n = 8k \]
   Since \( 8 = 2^2 + 2^2 + 0^2 + 0^2 \), \( 8k \) can be written as the sum of four squares.

2. **\( n \equiv 1 \mod 8 \):**
   \[ n = 8k + 1 \]
   Since \( 1 = 1^2 + 0^2 + 0^2 + 0^2 \), \( 8k + 1 \) can be written as the sum of four squares.

3. **\( n \equiv 2 \mod 8 \):**
   \[ n = 8k + 2 \]
   Since \( 2 = 1^2 + 1^2 + 0^2 + 0^2 \), \( 8k + 2 \) can be written as the sum of four squares.

4. **\( n \equiv 3 \mod 8 \):**
   \[ n = 8k + 3 \]
   Since \( 3 = 1^2 + 1^2 + 1^2 + 0^2 \), \( 8k + 3 \) can be written as the sum of four squares.

5. **\( n \equiv 4 \mod 8 \):**
   \[ n = 8k + 4 \]
   Since \( 4 = 2^2 + 0^2 + 0^2 + 0^2 \), \( 8k + 4 \) can be written as the sum of four squares.

6. **\( n \equiv 5 \mod 8 \):**
   \[ n = 8k + 5 \]
   Since \( 5 = 2^2 + 1^2 + 0^2 + 0^2 \), \( 8k + 5 \) can be written as the sum of four squares.

7. **\( n \equiv 6 \mod 8 \):**
   \[ n = 8k + 6 \]
   Since \( 6 = 2^2 + 1^2 + 1^2 + 0^2 \), \( 8k + 6 \) can be written as the sum of four squares.

8. **\( n \equiv 7 \mod 8 \):**
   \[ n = 8k + 7 \]
   Since \( 7 = 2^2 + 1^2 + 1^2 + 1^2 \), \( 8k + 7 \) can be written as the sum of four squares.

#### Conclusion
By considering all cases modulo 8 and using the identity for sums of four squares, we conclude that any natural number \( n \) can indeed be written as the sum of four squares. This completes the proof of Lagrange's Four Square Theorem.

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Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How does Lagrange's Four Square Theorem relate to other sum-of-squares theorems?
2. Can you prove that every number can be written as the sum of three triangular numbers?
3. What are the implications of Lagrange's Four Square Theorem in modern number theory?
4. Can the theorem be extended to sums of higher powers, such as cubes?
5. What are the historical contexts and developments leading to Lagrange's Four Square Theorem?
6. Are there efficient algorithms to find the four squares that sum to a given number?
7. How does the theorem apply in the context of quadratic forms and their classification?
8. Can you provide an example of expressing a large number as the sum of four squares?

**Tip:** Understanding modular arithmetic is crucial for many proofs in number theory, as it simplifies the analysis of numerical properties under different conditions.

Explore the proof of Lagrange's Four Square Theorem, which states that every natural number can be expressed as the sum of four integer squares. This detailed mathematical proof involves modular arithmetic, quadratic forms, and the properties of sums of squares. Ideal for advanced mathematics enthusiasts and students.

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