A three-term null vector for L_(T)+G_(T): half of a conjecture on -1 as a generalized eigenvalue of LCM matrices to GCD matrices

Abstract

For a finite set T of positive integers let L_(T)=(lcm(tᵢ,tⱼ)) and G_(T)=(gcd(tᵢ,tⱼ)) be the LCM and GCD matrices on T. Merikoski, Haukkanen, Sasaki and Tossavainen conjecture that, on T={1,…,n} with n>3, the number -1 is a generalized eigenvalue of L_(T) to G_(T) exactly when the binary expansion of n begins with 10; their evidence is a floating-point generalized eigensolver run to n=1000. Since -1 is a generalized eigenvalue precisely when det(L_(T)+G_(T))=0, this is a singularity statement about the integer matrix with entries lcm(i,j)+gcd(i,j). We prove the "if" half for every n, with no computation, by exhibiting a null vector: writing f(a,b)=lcm(a,b)+gcd(a,b), the three-term identity 2f(2ˢ,j)-3f(2ˢ⁺¹,j)+f(2ˢ⁺²,j)=0 holds for all 1 ≤ j<3 · 2ˢ⁺¹, so 2e_(2ˢ)-3e_(2ˢ⁺¹)+e_(2ˢ⁺²) annihilates L_(T)+G_(T) on the whole block 2ˢ⁺² ≤ n<3 · 2ˢ⁺¹, which is exactly the set of n whose binary expansion begins 10. The bound is sharp, and more is true: 2eₘ-3e₂ₘ+e₄ₘ annihilates L_(T)+G_(T) (for 4m ≤ n) if and only if m is a power of two and n<6m, which is why the answer to the conjecture is a condition on binary digits. The same identity is not about intervals at all, and yields an infinite family of index sets, of every size, on which -1 is a generalized eigenvalue: it suffices that T contain 2ˢ,2ˢ⁺¹,2ˢ⁺² and that 2ˢ⁺¹ be the only element of T of 2-adic valuation exactly s+1. This settles, for that family, the general-T case taken up in S4 of the source. The "only if" half of the conjecture remains open; we record an exact verification of the full biconditional for 4 ≤ n ≤ 47 inside the formal development and, outside it, for 2 ≤ n ≤ 1000. Two small errors in the published text are identified and corrected. Every theorem below is machine-checked in Lean 4; S[sec:verif] says exactly what the machine checks.

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