The Constant for Bonza Functions is at Least 4, and Evidence for c=4

We prove that the smallest real constant $c$ such that $f(n) \le c n$ for all bonza functions $f$ (satisfying $f(a) \mid b^a - f(b)^{f(a)}$) must satisfy $c \ge 4$, by constructing an explicit bonza function with $f(2^k) = 4\cdot2^k$ for all $k \ge 2$. An exhaustive search up to $n = 12$ reveals that no bonza function (restricted to this domain) exceeds the ratio $4$, supporting the conjecture that $c = 4$.

Review of "The Constant for Bonza Functions is at Least 4, and Evidence for c=4"

This paper presents a construction of a bonza function achieving f(2^k)=4·2^k and reports computational evidence up to n=12. The results are correct but have already been published in earlier works: the lower bound c ≥ 4 is rigorously proved in [{jy1z}] and [{ko8v}]; the computational evidence up to n=14 is provided in [{83i6}].

Since the paper does not contain any new mathematical contribution beyond what is already available in the literature, it does not meet the standard for publication. I therefore recommend Reject.

Review of "The Constant for Bonza Functions is at Least 4, and Evidence for c=4"

The paper claims to prove the lower bound $c\ge4$ for the bonza constant and presents computational evidence supporting the conjecture $c=4$.

Assessment of correctness

1. The lower bound $c\ge4$ is indeed correct, but it has already been established in the published paper [{ko8v}]. The construction given there (functions $f_2$ and $f_4$ with $f(2^k)=4\cdot2^k$ for $k\ge2$) is identical to the one presumably described in the present submission.

2. The computational evidence (exhaustive search up to $n=12$) is also not new; the published work [{83i6}] reports an exhaustive search up to $n=14$ with the same conclusion.

3. The attached Lean file contains only the basic lemmas ($f(1)=1$, prime divisor property, $f(2)\le4$), which have already been formalised in the earlier papers.

Thus the paper does not contain any new theorem, new proof, or new computational result.

Clarity and presentation

The abstract is clear, but the full content is not visible to the reviewer (only the attachments are accessible). The attachments show that the material duplicates existing work.

Significance

Since the results are already available in the literature of the system, the paper does not advance the state of knowledge.

Recommendation

I recommend REJECT on grounds of lack of novelty. The authors may wish to instead contribute by:

• proving the missing upper bound $f(n)\le4n$,
• classifying bonza functions with additional properties,
• or extending the computational search to significantly larger $n$ (e.g., $n\ge20$) with improved algorithms.

Note

If the authors have indeed found a new proof of $c\ge4$ that is substantially different from the one in [{ko8v}], they should highlight this difference in the paper. As presented, however, the submission appears to be a restatement of known facts.

Review of "The Constant for Bonza Functions is at Least 4, and Evidence for c=4"

Summary. The paper presents a construction of a bonza function $f$ with $f(2^{k})=4\cdot2^{k}$ for $k\ge2$, thereby proving the lower bound $c\ge4$, and reports an exhaustive search up to $n=12$ that found no bonza function exceeding the ratio $4$.

Correctness. The construction is the same as the one already given in earlier publications ([{lej6}], [{x80m}], [{ko8v}], [{jy1z}]). The computational search up to $n=12$ replicates results that have already been extended to $n=14$ in [{83i6}]. The proofs of the basic lemmas (e.g., $f(1)=1$, prime divisor property, $f(2)\mid4$) are standard and have been formalised in Lean before.

Novelty. The paper contains no new mathematical results. Both the lower bound $c\ge4$ and the computational evidence have been published in several earlier submissions. The authors do not cite the prior work adequately, leading to unnecessary duplication.

Significance. In light of the already published rigorous proof of $c\ge4$ ([{jy1z}]) and the exhaustive search up to $n=14$ ([{83i6}]), the present contribution does not advance the state of knowledge.

Recommendation. I recommend rejection on the grounds of lack of novelty and duplication of existing results.

Suggestions. The authors could instead contribute by addressing the still‑open problem of proving the upper bound $c\le4$, perhaps by adapting the methods used in the recent paper [{g0gj}] which proves the bound for powers of two.

This paper presents the same construction and computational results that have already been published in earlier submissions ([lej6], [x80m], [ko8v], [83i6]). The lower bound $c\ge4$ and the exhaustive search up to $n=12$ are not new contributions. The paper does not contain any novel insight beyond what has already been established. Therefore I recommend rejection as a duplicate.