{-1,1}-perfect numbers: a certified window to 10⁶, seven negative signs, and the least counterexample to Ross's Conjecture 17

Abstract

Ross calls n an S-perfect number of the first kind when the divisors of n strictly between 1 and n can be given coefficients from S so that 1+Σⱼλⱼdⱼ=n; for S={-1,1} this is a signed-divisor, or subset-sum, condition. He conjectured that every odd nonsquare n with σ(n) ≥ 2n is {-1,1}-perfect, with computational evidence to 10⁴; the conjecture was disproved by Campbell in July 2026, who produced counterexamples ap with a an odd square, σ(a) ≥ 2a, and p>σ(a) prime. Neither paper bounds the least counterexample. We prove, by exhibiting and re-checking a signing certificate for each of the 1989 admissible n, that every odd nonsquare n ≤ 10⁶ with σ(n) ≥ 2n is {-1,1}-perfect — a hundred times the published range — and, in the same sweep, that seven negative coefficients always suffice there, a refinement neither source considers and one that is not a theorem in waiting: the minimum number of negative coefficients needed is already 29 somewhere below 2.6 · 10⁸. We give a self-contained modulus certificate refuting n₀=253 277 325=3²5²7² · 22973, which is the least member of Campbell's construction and is about 6810 times smaller than the instance he prints; and we record an exhaustive search, run twice in independent implementations and outside the formal development, which finds n₀ to be the least counterexample of any shape and finds exactly 61 counterexamples below 2.6 · 10⁸, all of the form 11025p with p prime, 22973 ≤ p ≤ 23581. Ross's two printed lists are reproduced and completed over n ≤ 80. All statements below except those explicitly labelled as computations outside the development are machine-checked in Lean 4; S[sec:verif] says exactly what the machine checks. We also record plainly that the arXiv posting arXiv:2512.04417 of Ross's paper is a republication of the 2024 journal article and that the two texts differ: the journal prints the verification range 10⁴, the posting prints no range at all.

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