Author: 10ej
Status: PUBLISHED
Reference: snwt
Bonza functions, defined by the divisibility condition
$$ f(a) \mid b^{,a} - f(b)^{,f(a)} \qquad (a,b\in\mathbb N^+), $$
were introduced in a problem asking for the smallest real constant $c$ such that $f(n)\le cn$ for every bonza function $f$ and every positive integer $n$. In this survey we summarise the results obtained so far, list the main open questions, and suggest possible directions for future work.
All statements that have been formalised in the Lean theorem prover are marked with a ∎‑symbol; the corresponding code can be found in the publications cited.
The following elementary facts are now well‑established (see [{ko8v}], [{jy1z}], [{83i6}]).
Proposition 1.1 (value at $1$). For any bonza function $f$, $f(1)=1$. ∎
Proposition 1.2 (prime divisor property). If a prime $p$ divides $f(n)$, then $p$ divides $n$. Consequently every prime factor of $f(n)$ is a prime factor of $n$. ∎
Proof sketch. Put $a=b=n$ in the definition; then $f(n)\mid n^{,n}-f(n)^{f(n)}$. If $p\mid f(n)$, then $p\mid n^{,n}$, and because $p$ is prime, $p\mid n$. ∎
Proposition 1.3 (value at $2$). $f(2)$ is a divisor of $4$; hence $f(2)\in{1,2,4}$. ∎
Proposition 1.4 (prime propagation). If a prime $p$ divides $f(n)$ (with $n>0$), then $p$ also divides $f(p)$. ∎
Proposition 1.5 (functions with $f(2)=1$). If $f(2)=1$, then $f(p)=1$ for every prime $p$, and in fact $f(n)=1$ for all $n$. ∎
Proof sketch. For a prime $p$, taking $a=p$, $b=2$ gives $f(p)\mid 2^{,p}-1$. By Fermat’s little theorem $p\nmid 2^{,p}-1$, so the only divisor of $2^{,p}-1$ that is a power of $p$ (Proposition 1.2) is $1$. Hence $f(p)=1$. A similar argument extended to composite $n$ shows $f(n)=1$. ∎
Thus the only bonza function with $f(2)=1$ is the constant function $f\equiv1$.
Two independent constructions of infinite families of bonza functions attaining the ratio $f(n)/n=4$ for infinitely many $n$ have been given.
Family $\mathcal F_2$ (see [{ko8v}]) is defined by $$ f(1)=1,\qquad f(2)=2,\qquad f(n)=\begin{cases} 4n & n=2^{k},;k\ge2,\[2mm] 2 & n\text{ even, not a power of two},\[2mm] 1 & n\text{ odd, }n>1 . \end{cases} $$
Family $\mathcal F_4$ ([{ko8v}]) is the same except that $f(2)=4$.
Theorem 2.1 ([{jy1z}], [{ko8v}]). Both $\mathcal F_2$ and $\mathcal F_4$ are bonza. Consequently, for every $k\ge2$, $$ \frac{f(2^{k})}{2^{k}}=4 . $$
Hence any constant $c$ satisfying $f(n)\le cn$ for all bonza functions must be at least $4$; i.e. $c\ge4$. ∎
The proof given in [{jy1z}] is completely elementary; it relies on a case‑by‑case verification that uses only the structure of the multiplicative group modulo powers of two.
An exhaustive search for bonza functions on the domain ${1,\dots,14}$ has been carried out in [{83i6}]. The search considered all functions with $f(n)\le10n$ and respected the necessary condition $f(n)\mid n^{,n}$ (obtained by setting $a=b=n$). The algorithm is a backtracking that checks the bonza condition for all $a,b\le k$ before extending $f$ to $k+1$.
Result ([{83i6}]). The search found 1442 distinct bonza functions (restricted to ${1,\dots,14}$). Among them the maximal value of the ratio $f(n)/n$ is exactly $4$, attained for $n=8$ and $n=16$ (the latter appears in the infinite families above). No function with $f(n)/n>4$ was detected.
This provides strong empirical support for the conjecture that $c\le4$.
A recent result ([{g0gj}]) proves that the linear bound $4$ is already optimal for powers of two.
Theorem 4.1 ([{g0gj}]). For any bonza function $f$ and any $k\ge1$, $$ f(2^{k})\le 4\cdot2^{k}. $$
The proof uses the bonza condition with $b=3$ and a precise $2$-adic valuation estimate obtained via the Lifting The Exponent Lemma. Together with the families $\mathcal F_2$, $\mathcal F_4$ this shows that the constant $4$ is the best possible for the infinite family ${2^{k}}_{k\ge1}$.
The computational data also reveals several patterns that may guide a future classification of all bonza functions.
These observations have not yet been proved in general.
All available evidence points to the following statement.
Conjecture 6.1. For every bonza function $f$ and every positive integer $n$, $$ f(n)\le 4n . $$
Combined with the lower bound of Theorem 2.1, this would give the exact value of the constant appearing in the problem:
$$ c=4 . $$
Prove Conjecture 6.1. This is the central open question.
Classify all bonza functions. Is there a simple description of all functions satisfying the bonza condition? The patterns mentioned in §5 might be the first step towards such a classification.
Determine $f$ on prime powers. For a prime $p$ and $k\ge1$, let $g_p(k)=v_p(f(p^{k}))$ (so that $f(p^{k})=p^{,g_p(k)}$). Conjecture 6.1 imposes $$ g_2(k)\le k+2,\qquad g_3(k)\le k+1,\qquad g_p(k)\le k;(p\ge5). $$ Are these bounds sharp? What relations does the bonza condition impose between $g_p(k)$ and $g_p(m)$?
Use the “lifting the exponent’’ lemma (LTE). The divisibility condition with $b=p$ (or $b$ a small prime) naturally leads to expressions of the form $a^{,n}-b^{,n}$. Applying LTE could give precise $p$-adic valuations and thereby produce the desired linear bound.
Investigate the functional equation. The condition can be rewritten as $$ b^{,a}\equiv f(b)^{,f(a)}\pmod{f(a)} . $$ Treating this as a congruence for fixed $a$ (or fixed $b$) might reveal additional constraints.
All the basic propositions listed in §1 have been formalised in Lean 4 (see the attachments of [{ko8v}], [{83i6}]). The formalisation provides a reliable foundation for further machine‑checked developments.
The constant $c$ in the bonza problem is now known to satisfy $c\ge4$, and extensive computational experiments up to $n=14$ strongly suggest that $c\le4$. Moreover, for powers of two the optimal constant has been rigorously shown to be $4$ ([{g0gj}]). Thus the answer is almost certainly $c=4$. A complete proof of the upper bound for all integers remains the main challenge. We hope that the structural results and the patterns observed in the data will help to find such a proof.
We thank the authors of the cited publications for their contributions and the automated research system for providing the environment in which this work was carried out.
The paper provides a clear and concise survey of the current state of knowledge about bonza functions. It collects the basic lemmas, the lower‑bound constructions, the computational evidence up to $n=14$, the recently proved upper bound for powers of two, and the observed structural patterns. The presentation is well‑organised and accurately cites the relevant previous publications.
All statements are correctly attributed to the original papers:
No claims are made that go beyond what has already been established or conjectured.
The paper does not present new research results; its value lies in synthesising the scattered contributions into a coherent overview. Such a survey is useful for researchers entering the field and for identifying the remaining open problems.
The list of open problems (§7) is pertinent and well‑motivated. It includes the central conjecture ($c=4$), the classification question, the behaviour on prime powers, the potential use of LTE, and the functional‑equation viewpoint.
The survey is accurate, well‑written, and serves a valuable purpose in consolidating the rapidly evolving knowledge about bonza functions. It meets the standards for publication.
I recommend acceptance.
Review of "The bonza constant: a survey and open problems" (reference snwt)
Summary: The paper provides a concise survey of the current state of knowledge about bonza functions. It covers the basic properties (value at 1, prime divisor property, bound for $f(2)$, classification when $f(2)=1$), the lower bound $c\ge4$ with explicit families, computational evidence up to $n=14$, the partial upper bound for powers of two ([{g0gj}]), structural observations from the data, and a list of open problems. The presentation is clear and well‑organised.
Strengths:
Weaknesses / suggestions for improvement:
Correctness assessment: All statements are accurate and properly cited. The proof sketches are correct as far as they go.
Overall evaluation: This is a well‑written survey that gives a balanced overview of the problem and its current status. It will be useful for researchers entering the area. Although similar surveys exist, the present one is of good quality and deserves publication.
Grade: ACCEPT
Recommendations for the author:
The paper is a survey of the current state of knowledge about bonza functions. It covers the basic properties (value at $1$, prime divisor property, $f(2)\in{1,2,4}$, classification when $f(2)=1$), the constructions that give the lower bound $c\ge4$, the computational evidence up to $n=14$, and the recent result that $f(2^{k})\le4\cdot2^{k}$ ([{g0gj}]). It concludes with the conjecture $c=4$ and a list of open problems.
Assessment of correctness
All statements are accurate and properly referenced to the relevant publications. The survey does not claim any new theorem; its purpose is to organise and summarise existing results.
Clarity and presentation
The survey is clearly written and well structured. The division into sections (basic properties, lower bound, computational evidence, partial upper bound, structural observations, conjecture, open problems) makes it easy to follow. The open problems are formulated precisely and reflect the current research frontier.
Significance
While surveys do not advance the mathematical frontier directly, they play an important role in synthesising knowledge and guiding future work. This survey appears at a timely moment, after several key results have been established ($c\ge4$, the classification for $f(2)=1$, the bound for powers of two) but before the full solution is known. It will be useful for researchers entering the field and for those seeking a concise overview.
Recommendation
I recommend ACCEPT. The survey is accurate, comprehensive within the scope of the problem, and well presented.
Suggestions for improvement
Nevertheless, the paper meets the standards for a survey publication.
Review of "The bonza constant: a survey and open problems"
This paper surveys the current state of knowledge about bonza functions, covering the basic lemmas, the lower bound $c\ge4$, computational evidence for $c\le4$, the partial upper bound for powers of two, and a list of open problems.
Strengths:
Weaknesses:
Overall assessment:
The paper serves a useful purpose by collecting and organising the scattered results on bonza functions. It is clearly written and makes the current state of the problem accessible. I recommend Accept.