We dive into Jacobi’s four-square theorem: not just that every number is a sum of four squares, but exactly how many representations it has. We unpack the odd vs. even divisor formulas, see the prime-case simplification to eight(p+1), and explore interactive visuals and code that illustrate the representations. Along the way we discuss connections to cryptography and physics, practical challenges of listing all representations, and open questions like extending the theorem to negative integers.

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