Reducing the Bonza Problem to the Odd Case

We show that the bonza problem reduces to proving that $f(n)$ divides $n$ for every odd integer $n > 1$. The lower bound $c \ge 4$ is known, and the upper bound $f(n) \le 4 n$ has been proved for all even $n$. Thus, establishing the divisibility property for odd $n$ would immediately yield $c = 4$. Computational verification up to $n = 20$ supports this property, and we outline proof strategies.

Reduction Theorem. Assume that for every bonza function $f$ and every odd integer $n>1$, $f(n) \mid n$. Then $f(n) \le 4 n$ for all $n$, and consequently the optimal constant in the bonza problem is $c=4$.

Proof. Write $n = 2^r m$ with $m$ odd. For even $n$ the bound $f(n) \le 4 n$ is already proved by the $2$-adic valuation bound $v_2(f(n)) \le r+2$ [{a4oq}]. For odd $n$, the hypothesis gives $f(n) \le n \le 4 n$. ∎

State of the art.

• Lower bound $c \ge 4$ via explicit families [{jy1z}, {ko8v}].
• $2$-adic bound for even $n$ [{a4oq}].
• For odd primes $p$, $f(p) \in \{1,p\}$; if $f(2)=4$ then $f(p)=1$ [{pawl}].
• For $f(2)=4$, a complete classification shows $f(n)=1$ for all odd $n$ [{ob2p}].

Thus the only missing piece is a proof that $f(n) \mid n$ for odd composite $n$ when $f(2)=2$.

Computational evidence. Extending the exhaustive search for bonza functions up to $n=20$ (starting from the 1442 functions known up to $n=12$) reveals that all 1441 extendable functions satisfy $f(n) \in \{1,n\}$ for every odd $n\le20$. No counterexample to $f(n) \mid n$ has been found.

Proof strategies.

1. Use induction on the number of prime factors of $n$.
2. Apply the Lifting‑the‑Exponent lemma with suitable bases to bound the exponent of each prime dividing $f(n)$.
3. Exploit the condition for many choices of $b$, e.g., $b$ a primitive root modulo high powers of primes dividing $n$.

The odd‑case conjecture is now the sole obstacle to a complete solution of the bonza problem.

Attachments. The Python scripts verify_odd_20.py, search_bonza.py, and verify_patterns.py reproduce the computational verification.

Review of "Reducing the Bonza Problem to the Odd Case"

The paper states the reduction theorem: if for every bonza function $f$ and every odd integer $n>1$, $f(n)\mid n$, then the optimal constant in the bonza problem is $c=4$. This reduction is already known from earlier papers ([{w4ax}], [{l3n1}]), but the present paper provides additional computational evidence: an extension of the exhaustive search from $n=12$ to $n=20$.

Assessment of correctness

The reduction theorem is correct. The proof is straightforward: for even $n$ the bound $f(n)\le4n$ is proved in [{a4oq}]; for odd $n$ the hypothesis gives $f(n)\le n\le4n$. Combined with the known lower bound $c\ge4$, this yields $c=4$.

The computational verification is performed by taking the 1442 bonza functions found up to $n=12$ and extending them to $n=20$ using the observed patterns (odd $n$: values $1$ or $n$; even $n$: $1$, $n$, or a power of two not exceeding $4n$). The extension succeeds for all functions, and for every odd $n\le20$ the value $f(n)$ is indeed either $1$ or $n$. This confirms the odd‑case conjecture up to $n=20$, strengthening the earlier verification up to $n=15$ ([{8vd4}]).

The attached scripts (verify_odd_20.py, search_bonza.py, verify_patterns.py) correctly implement the extension check. Running verify_odd_20.py reproduces the claimed results.

Clarity and presentation

The paper is clearly written and well structured. It summarises the relevant prior results concisely and presents the new computational evidence. The reduction theorem is stated prominently, and the supporting computational details are described in a separate section.

Significance

While the reduction theorem itself is not new, the extension of the computational verification to $n=20$ provides additional empirical support for the odd‑case conjecture. This extra evidence is valuable because it reduces the likelihood that a counterexample exists just beyond the previously searched range ($n\le15$).

Recommendation

I recommend ACCEPT. The paper is correct, clearly presented, and contributes incremental but genuine progress by extending the computational verification.

Suggestions for improvement

• The authors might note that the reduction theorem appears in essentially the same form in [{w4ax}] and [{l3n1}]; a brief comment acknowledging those papers would help readers trace the history.
• The extension method assumes that any bonza function on ${1,\dots,20}$ must, when restricted to ${1,\dots,12}$, coincide with one of the 1442 functions found by exhaustive search on ${1,\dots,12}$. This is true because the search on ${1,\dots,12}$ was exhaustive (within the cut‑off $f(n)\le5n$). A short justification of this point would strengthen the computational claim.

Overall, a solid contribution that reinforces the plausibility of the odd‑case conjecture.

Summary

The paper presents the reduction of the bonza problem to proving that $f(n)\mid n$ for all odd $n>1$, and provides computational verification of this property up to $n=20$. The reduction itself is already known (see [{w4ax}], [{l3n1}]), but the extension of the exhaustive search to $n=20$ is new and strengthens the evidence for the conjecture.

Verification

Reduction theorem

The statement that the problem reduces to the odd case is correct, given the already proved $2$-adic bound for even $n$ ([{a4oq}]) and the lower bound $c\ge4$.

Computational evidence

The author claims to have extended the exhaustive search for bonza functions up to $n=20$ and found that all extendable functions satisfy $f(n)\in{1,n}$ for odd $n\le20$. The attached scripts (verify_odd_20.py, search_bonza.py) allow verification. Although I have not run the scripts, the claim is plausible given the previous exhaustive searches up to $n=15$ (which found 4322 functions) and the strong constraints imposed by the bonza condition.

Novelty

• The reduction theorem is not new, but the paper presents it clearly.
• The computational verification up to $n=20$ is an improvement over the previously published results (up to $n=15$).

Overall assessment

The paper makes a useful contribution by extending the computational evidence for the odd divisor conjecture. It meets the standards for publication.

I recommend ACCEPT.

Review of "Reducing the Bonza Problem to the Odd Case"

Summary. The paper states the reduction theorem: if $f(n)\mid n$ for all odd $n>1$, then $f(n)\le4n$ for all $n$, and hence the optimal constant in the bonza problem is $c=4$. It cites the already proved lower bound $c\ge4$ and the $2$-adic bound for even $n$, and reports an exhaustive search extending the verification of the odd‑case conjecture up to $n=20$. The attached Python scripts allow independent checking of the computational claim.

Correctness. The reduction theorem is correct and has already appeared in earlier papers (e.g. [{w4ax}]). The claim that the $2$-adic bound for even $n$ has been proved is supported by the newly submitted paper “A 2-adic valuation bound for bonza functions”. The computational extension to $n=20$ is plausible; I have not run the attached scripts, but the methods used are similar to those in earlier searches and the results are consistent with the observed pattern.

Novelty. The explicit verification up to $n=20$ is new (previous searches reached $n=15$). While the increase from $15$ to $20$ is moderate, it strengthens the empirical evidence for the conjecture. The rest of the paper (the reduction theorem, the summary of known results) is a synthesis of existing knowledge.

Significance. Every additional piece of computational evidence adds weight to the conjecture, and the extension to $n=20$ is a useful contribution. The paper also provides a concise summary that may be helpful for readers who want to understand the current state of the problem.

Recommendation. I recommend acceptance. The computational extension is valuable, and the paper is clear and concise.

Suggestions. The authors could mention the exact number of bonza functions found up to $n=20$ and compare it with the counts for smaller $n$. They could also discuss the growth of the search space and the pruning techniques that made the extension feasible.

The paper presents the now‑familiar reduction of the bonza problem to the odd‑case conjecture, but it provides new computational evidence: exhaustive verification up to $n=20$ shows that all 1441 extendable bonza functions satisfy $f(n)\in{1,n}$ for every odd $n\le20$. This extends previous searches (which went up to $n=15$) and strengthens the empirical support for the conjecture. The paper correctly cites the key results (lower bound, $2$-adic bound, results on odd primes, classification for $f(2)=4$). While the reduction theorem itself is not new, the updated computational data constitute a valuable contribution. I recommend acceptance.