A Paley colouring settles D_(Qₚ,1): the quadratic-residue weighted Davenport constant is 4 for every prime p ≡ 1 (mod 4)

Abstract

For an odd prime p let Qₚ be the set of nonzero quadratic residues modulo p and let 1={1}. A sequence (x₁,…,xₖ) in Zn(p)=ℤ/pℤ is a (Qₚ,1)-weighted zero-sum sequence if there are a₁,…,aₖ ∈ Qₚ with a₁x₁+…+aₖxₖ=0 and a₁+…+aₖ=0, and D_(Qₚ,1)(p) is the least k such that every length-k sequence in Zn(p) has such a subsequence. Paul and Paul, who introduced this doubly-weighted constant, proved D_(Qₚ,1) ∈ {4,5} for every odd p and D_(Qₚ,1)=5 for p ≡ 3 (mod 4), and wrote of the remaining case: "we believe that D_(Qₚ,1)=4 … However, despite our best efforts, we have been unable to show this". We prove it. The mechanism is that for p ≡ 1 (mod 4) the element -1 is a square, so the quadratic character χ(x-y) is a well-defined 2-colouring of the edges of the complete graph on Zn(p) — the Paley colouring — and each of three configurations forces a weighted zero-sum subsequence: a repeated term, a monochromatic triangle, and a monochromatic perfect matching, the last through the weight vector (1,-1,-u,u). Since every 2-colouring of the edges of K₄ contains a monochromatic triangle or a monochromatic perfect matching, four terms always suffice. With the authors' own theorem for p ≡ 3 (mod 4) the value of D_(Qₚ,1)(p) is now known at every odd prime, and their conditional consequence becomes unconditional: D_(Qₚ,B)(p)=4 for p ≡ 1 (mod 4) and every nonempty B with 0 ∉ B. Every theorem below is machine-checked in Lean 4; S[sec:verif] says exactly what the machine checks. An independent exhaustive computation, outside the formal development, confirms the value at all 168 odd primes p ≤ 1009.

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