In solving the Legendre's four square theorem, provide the formula  if product of two numbers expressed as sums of four squares, the product can be expressed as sums of four squares

The Legendre's four-square theorem states that any natural number can be expressed as the sum of four squares. However, there's also an important result related to the product of two numbers, each of which is a sum of four squares.

If \(a\) and \(b\) are two numbers that can each be expressed as the sum of four squares, say:
\[
a = a_1^2 + a_2^2 + a_3^2 + a_4^2
\]
\[
b = b_1^2 + b_2^2 + b_3^2 + b_4^2
\]

then their product \(ab\) can also be expressed as the sum of four squares, and the formula is given by:

\[
ab = (a_1b_1 - a_2b_2 - a_3b_3 - a_4b_4)^2 + (a_1b_2 + a_2b_1 + a_3b_4 - a_4b_3)^2 + (a_1b_3 - a_2b_4 + a_3b_1 + a_4b_2)^2 + (a_1b_4 + a_2b_3 - a_3b_2 + a_4b_1)^2
\]

This formula shows that the product of two sums of four squares is itself a sum of four squares.

Would you like more details or have any questions?

Here are some related questions:
1. What is the significance of the Legendre's four-square theorem in number theory?
2. How can the four-square theorem be applied in solving Diophantine equations?
3. Are there other formulas to express the product of sums of squares?
4. How does the four-square theorem relate to quaternions?
5. Can you provide an example calculation using the above formula?

**Tip:** The four-square theorem is a special case of the more general theorem for sums of squares in various dimensions, connected to algebraic structures like quaternions and octonions.

Explore Legendre's four-square theorem and its application in number theory. Learn how the product of two numbers, each expressible as sums of four squares, can also be expressed in the same form. Includes detailed formula and examples.

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