OEIS A000236: Adjacent Quadratic Residues

Join us for a focused look at adjacent quadratic residues in modular arithmetic. We classify consecutive pairs (k, k+1) modulo a prime p using Legendre symbols into four sets: RR (both residues), RN (residue then non-residue), NR (non-residue then residue), and NN (both non-residues). Let the counts be alpha_{RR}, alpha_{RN}, alpha_{NR}, alpha_{NN}. The pattern depends crucially on p mod 4: if p ≡ 1 (mod 4), RR = (p−5)/4 and RN = NR = NN = (p−1)/4; if p ≡ 3 (mod 4), RN = (p+1)/4 and RR = NR = NN = (p−3)/4. We illustrate with p = 17 and p = 19, connect to Gauss’s classical work on residues, and touch on related questions like solvability of x^4 ≡ 2 (mod p). We’ll also point to OEIS A000236 for deeper context and further patterns in quadratic residues.

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