We unpack A000089, the OEIS entry counting solutions to x^2 ≡ -1 (mod n). We'll explore its multiplicative structure, the no-solution case when n is divisible by 4, and why primes p ≡ 1 (mod 4) yield exactly two solutions. Along the way we touch on Legendre symbols, quadratic residues, and the deeper number-theory connections that make this tiny sequence surprisingly rich.

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