Fow which type of numbers, the numbers cannot be expressed as four squares

All positive integers can be expressed as the sum of four squares, as guaranteed by Lagrange's Four Square Theorem. However, if you're asking about numbers that cannot be expressed as a sum of fewer than four squares, this depends on specific types of numbers.

### Numbers That Cannot Be Expressed as a Sum of Fewer Than Four Squares:
1. **Numbers of the form \( 4^k(8m + 7) \):** These numbers cannot be expressed as the sum of three squares, and therefore, by the theorem, require four squares for their representation.

### Summary:
- **All positive integers** can be expressed as the sum of four squares.
- **Numbers of the form \( 4^k(8m + 7) \)** require exactly four squares and cannot be expressed as the sum of fewer than four squares.

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Here are some related questions:
1. Can you give examples of numbers that require exactly four squares?
2. How does the proof of Lagrange's Four Square Theorem work?
3. What is the significance of numbers in the form \( 4^k(8m + 7) \) in number theory?
4. Are there generalizations of the Four Square Theorem to more squares?
5. How do quadratic forms relate to the representation of numbers as sums of squares?

**Tip:** Numbers of the form \( 4^k(8m + 7) \) being excluded from representations with three squares is linked to properties of quadratic forms and modular arithmetic.

Explore the concept of numbers that cannot be expressed as fewer than four squares, including the theorem by Lagrange. Learn about specific forms of numbers that require exactly four squares for representation.

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