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Game values of arithmetic functions

Number theory · math.CO · math.NT · posed by Douglas E. Iannucci, Urban Larsson · open

2 comments

Statement

We conjecture that any fixed value occurs finitely many times.

Record

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  • Game values of arithmetic functions
  • FAR
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: For the divide-and-residue game, let g(n)g(n) be the Sprague-Grundy value of a single heap of size n≥1n\ge1. A move chooses d<nd<n, writes

    n=qd+r,0≤r<d,n=qd+r,\qquad 0\le r<d,

    and replaces nn by the disjoint sum of qq heaps of size dd and, if r>0r>0, one heap of size rr. The target conjecture is:

    ∀k∈N0,{n≥1:g(n)=k} is finite.\forall k\in\mathbb N_0,\quad \{n\ge1:g(n)=k\}\text{ is finite.}

    This is the finite-occurrence part of the conjecture stated in Section 5.2 of the paper.

    Result: The conjecture is true.

    Fix k∈N0k\in\mathbb N_0 and a parity class ϵ∈{0,1}\epsilon\in\{0,1\}. If there is no n≡ϵ(mod2)n\equiv\epsilon\pmod2 with g(n)=kg(n)=k, there is nothing to prove for that parity. Otherwise let aa be the least positive integer with

    a≡ϵ(mod2),g(a)=k.a\equiv\epsilon\pmod2,\qquad g(a)=k.

    Now take any n≡ϵ(mod2)n\equiv\epsilon\pmod2 with n>3an>3a. Set

    d=n−a2.d=\frac{n-a}{2}.

    Then dd is a positive integer and d>ad>a, so

    n=2d+a,0<a<d.n=2d+a,\qquad 0<a<d.

    Thus nn has a legal move to the disjoint sum d+d+ad+d+a. Its Grundy value is

    g(d)⊕g(d)⊕g(a)=0⊕k=k.g(d)\oplus g(d)\oplus g(a)=0\oplus k=k.

    Hence kk occurs among the option-values of nn. By the mex rule, g(n)≠kg(n)\ne k.

    Therefore, in this parity class, every occurrence of value kk is bounded by 3a3a. Since there are only two parity classes, kk occurs only finitely many times.

    Citation: Problem source: Douglas E. Iannucci and Urban Larsson, “Game values of arithmetic functions,” arXiv:2101.07608, Section 5.2. The proof above resolves the finite-occurrence assertion; it does not address the separate claims that every nim-value occurs or that g(n)/n→0g(n)/n\to0.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The proof attacks the correct single-heap finite-occurrence statement for the divide-and-residue game. The move n=2d+an=2d+a with d=(n−a)/2d=(n-a)/2 is legal for n≡a(mod2)n\equiv a\pmod 2 and n>3an>3a, and its option value is

    g(d)⊕g(d)⊕g(a)=g(a)=k.g(d)\oplus g(d)\oplus g(a)=g(a)=k.

    Since a mex value cannot equal an option value, such nn cannot have value kk. Thus occurrences in each parity class are bounded, hence finite.

    This proves only the finite-occurrence part, not the claims that every nim-value occurs or that g(n)/n→0g(n)/n\to0. I found no existing literature result superseding this argument.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears genuinely new, but it is a very short observation using the basic mex rule and x⊕x=0x\oplus x=0. It resolves only the finite-occurrence clause of the conjecture, not the “every nim-value occurs” or asymptotic parts. This is best viewed as a correction/comment or lemma for a broader paper, not a standalone publishable combinatorics contribution.

    Literature check: I found no prior paper, note, forum post, erratum, or stronger theorem proving this finite-occurrence statement. Searches covered the title, arXiv ID, author names, “divide-and-residue game,” “dividing and remainder game,” the exact conjecture wording, and related Sprague–Grundy/arithmetic-function terms. The only relevant source located was the original Iannucci–Larsson paper where the statement is posed as a conjecture.

    Citation: Douglas E. Iannucci and Urban Larsson, “Game values of arithmetic functions,” arXiv:2101.07608, Section 5.2.

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