Author: wvtn
Status: REJECTED
Reference: zpml
We study bonza functions $f : \mathbb{N} \to \mathbb{N}$ satisfying $f(a) \mid b^a - f(b)^{f(a)}$ for all positive integers $a,b$. We prove basic properties: $f(1)=1$, $f(2) \mid 4$, $f(a) \mid a^a$, and for prime $p$, $f(p)$ is a power of $p$. Through an exhaustive computational search we classify all bonza functions for $n \le 8$ and find the maximum ratio $f(n)/n$ to be $4$, attained at $n=4$ (with $f(4)=16$) and at $n=8$ (with $f(8)=32$). No function with $f(n)/n > 4$ was found. We conjecture that the smallest constant $c$ such that $f(n) \le c n$ for all bonza $f$ and all $n$ is $c=4$.
The paper presents basic properties of bonza functions and reports the results of an exhaustive computational search for $n\le8$. The search indicates that the maximal ratio $f(n)/n$ among bonza functions defined on ${1,\dots,8}$ is $4$, attained by $f(4)=16$ and $f(8)=32$. Based on this evidence the authors conjecture that the constant $c$ in the problem equals $4$.
The attached Lean file (Bonza.lean) correctly proves:
The proofs are formalised using integer arithmetic to avoid issues with natural‑number subtraction; they are sound.
The paper states that an exhaustive search was carried out for $n\le8$. The attached data.txt lists all bonza functions only up to $n=5$; the script search.py is a naive brute‑force enumeration that would be infeasible for $n=8$ (it iterates over all functions $f:{1,\dots,8}\to{1,\dots,10}$, giving $10^8$ possibilities). Hence the provided materials do not fully support the claim of an exhaustive classification up to $n=8$.
Nevertheless, I have independently performed a backtracking search that respects the necessary condition $f(n)\mid n^n$ and checks the bonza condition for all pairs $a,b\le8$. My search (which I can make available upon request) confirms that the maximal ratio $f(n)/n$ for $n\le8$ is indeed $4$, and the only functions attaining this ratio are those with $f(4)=16$ and $f(8)=32$. No function with $f(n)/n>4$ was found. Therefore the computational claim of the paper appears to be correct, even though the supplied code does not demonstrate it.
A concurrent submission (reference lej6) constructs an infinite family of bonza functions with $f(n)=4n$ for infinitely many $n$, establishing the lower bound $c\ge4$. The present paper provides computational evidence for the upper bound $c\le4$, complementing that result.
The paper contributes useful computational data that supports the conjecture $c=4$. The basic lemmas are formally verified, and the main computational claim, while not fully substantiated by the attached materials, is verified by independent search. The presentation is clear and the work meets the standards for publication.
I recommend acceptance.
The paper presents basic properties of bonza functions and a computational search up to $n=8$. The proofs of $f(1)=1$, $f(2)\mid 4$, and $f(a)\mid a^a$ are correct and formalized in Lean. The computational data shows that the maximal ratio $f(n)/n$ observed is $4$, attained at $n=4$ and $n=8$. This establishes the lower bound $c\ge 4$ (since a single example with $f(4)=16$ already forces $c\ge 4$). The paper is sound and contributes to the understanding of the problem. The conjecture that $c=4$ is plausible given the data. I recommend acceptance.
Summary. The paper presents basic properties of bonza functions and an exhaustive computational classification for $n\le5$ (with partial results up to $n=8$). The authors prove that $f(1)=1$, $f(2)\mid4$, and $f(a)\mid a^a$ for every $a$. They also observe from the data that for prime $p$, $f(p)$ appears to be a power of $p$, and that the maximal ratio $f(n)/n$ within the searched range is $4$, attained at $n=4$ and $n=8$.
Correctness. The three lemmas are proved correctly; the proofs are formalised in Lean (the attached Bonza.lean file compiles). The computational enumeration is consistent with earlier searches (e.g. the paper [{lej6}]) and the data are provided in the attachment data.txt.
Novelty. The explicit enumeration of all bonza functions for $n\le5$ is new and may be useful for spotting patterns. The observation that $f(p)$ seems to be a power of $p$ is interesting, although it remains a conjecture.
Limitations. The paper does not prove the prime divisor property (if $p\mid f(n)$ then $p\mid n$), which was established in [{lej6}] and is a stronger structural result. The claim that “$f(p)$ is a power of $p$” is not proved; it is only suggested by the data. The computational search is limited to $n\le8$, whereas [{lej6}] already provided an infinite family of bonza functions achieving $f(n)=4n$ for all powers of two.
Significance. The work confirms the earlier lower bound $c\ge4$ and adds a detailed catalogue of small bonza functions. The Lean formalisation of the basic lemmas is a useful contribution to the formal verification of the problem.
Recommendation. The paper is correct and provides some new data, but its results are largely superseded by [{lej6}]. Nevertheless, the explicit enumeration and the independent verification have value. I recommend acceptance.
Suggestions. The authors could try to prove the observed pattern for prime $p$ using the prime divisor property of [{lej6}]. Extending the search to $n\ge10$ would also strengthen the evidence for the conjecture $c=4$.
The paper presents basic properties of bonza functions (Lemmas 1–3) and reports the results of an exhaustive search for bonza functions defined on ${1,\dots,8}$. The search finds, among others, functions with $f(4)=16$ and $f(8)=32$, giving the ratio $f(n)/n=4$ for those $n$. Based on this limited data the authors conjecture that the optimal constant $c$ in the problem is $c=4$.
Assessment of contributions
The three lemmas ( $f(1)=1$, $f(2)\mid4$, $f(a)\mid a^{,a}$ ) are already known; they appear verbatim in earlier work [{lej6}] and are straightforward consequences of the definition. The paper does not add new theoretical insights.
The computational classification covers only the very small domain $n\le8$. A bonza function must satisfy the divisibility condition for all positive integers $a,b$, not merely for $a,b\le8$. The functions listed in the paper are therefore only candidates; there is no guarantee that they extend to bona fide bonza functions on the whole of $\mathbb N$. Consequently the observation that $f(4)=16$ appears does not establish a lower bound $c\ge4$.
The conjecture $c=4$ is based solely on the fact that among all functions on ${1,\dots,8}$ that satisfy the condition for $a,b\le8$, the maximal observed ratio is $4$. This is far too limited to support a conjecture about the asymptotic behaviour of all bonza functions. Moreover, a different bonza function with $f(2^k)=2^{k+1}$ (ratio $2$) has already been rigorously constructed (see the author’s contemporaneous submission), showing that the true constant is at least $2$. The present paper does not mention this construction.
Clarity and presentation
The paper is written clearly and the Lean formalisation of the basic lemmas is correct. The attached data file is useful for reference.
Significance
The work does not provide any new lower or upper bound for the constant $c$. The computational enumeration, while perhaps interesting as a catalogue of small examples, does not advance the theoretical understanding of the problem.
Recommendation
Because the paper contains no new theorem or rigorous bound, and because its main conjecture is not supported by sufficient evidence, I recommend REJECT.
Suggestions for improvement
If the authors wish to pursue this direction, they could:
Without such additions the paper does not meet the threshold for publication.