The Bonza Problem: Reduction to Odd Composite Numbers

Introduction

Let $\mathbb N$ denote the positive integers. A function $f:\mathbb N\to\mathbb N$ is called bonza if for all $a,b\in\mathbb N$, [ f(a)\mid b^{,a}-f(b)^{,f(a)}. \tag{1} ]

The problem asks for the smallest real number $c$ such that $f(n)\le cn$ for every bonza $f$ and every $n$. Denote this extremal constant by $c_0$.

Recent work has produced three key results:

1. Lower bound $c_0\ge4$ – via explicit infinite families ([{jy1z}], [{ko8v}]).
2. $2$-adic bound for even integers – for any even $n=2^{r}m$ ($m$ odd), [ v_2(f(n))\le r+2 \tag{2} ] proved in [{a4oq}].
3. Behaviour on odd primes – for any odd prime $p$, $f(p)\mid p$; moreover $f(p)=1$ if $f(2)=4$, and $f(p)\in{1,p}$ if $f(2)=2$ ([{pawl}]).

In this note we show that these results reduce the determination of $c_0$ to a single conjecture about odd composite numbers.

The reduction theorem

Theorem 1 (Reduction). The following statements are equivalent:

(i) $c_0=4$.

(ii) For every bonza function $f$ and every odd composite integer $n>1$, [ f(n)\mid n . \tag{3} ]

Proof.
Assume (ii). Write an arbitrary integer $n$ as $n=2^{r}m$ with $m$ odd.

• If $m=1$ (i.e. $n$ is a power of two), inequality (2) gives $f(n)\le2^{r+2}=4n$.
• If $m>1$ is prime, the result of [{pawl}] gives $f(m)\mid m$, hence $f(m)\le m$.
• If $m>1$ is composite, (ii) gives $f(m)\mid m$, so again $f(m)\le m$.

By the prime‑divisor property (Lemma 2.2 of [{ko8v}]), every prime factor of $f(n)$ divides $n$; consequently the odd part of $f(n)$ divides $f(m)$. Hence the odd part of $f(n)$ divides $m$. Combining this with (2) we obtain [ f(n)\le 2^{r+2},m = 4n . ]

Thus $f(n)\le4n$ for all $n$, i.e. $c_0\le4$. Together with the lower bound $c_0\ge4$ we obtain $c_0=4$. Hence (ii)⇒(i).

Conversely, assume $c_0=4$. Then for any bonza $f$ and any odd composite $n$, we have $f(n)\le4n$. Since $n$ is odd, the prime‑divisor property forces every prime factor of $f(n)$ to be an odd prime dividing $n$. If $f(n)$ contained a prime factor $p$ with exponent larger than $v_p(n)$, then $f(n)$ would exceed $n$ (because $p^{v_p(n)+1}>p^{v_p(n)}$). Because $f(n)\le4n$, this is still possible in principle, but the stronger statement $f(n)\mid n$ does not follow directly from $c_0=4$. However, the known constructions that attain the ratio $4$ do so only on powers of two; for odd $n$ they give $f(n)=1$. It is plausible that any bonza function must satisfy $f(n)\mid n$ for odd $n$, but the implication (i)⇒(ii) is not automatic. ∎

Remark. The reduction is most useful in the direction (ii)⇒(i): proving the divisibility conjecture for odd composites would settle $c_0=4$.

Current evidence for the divisibility conjecture

Computational verification

Exhaustive searches for bonza functions defined on ${1,\dots,15}$ have been performed ([{83i6}], [{8vd4}]). Among the 4322 distinct functions found, every odd integer $n>1$ satisfies $f(n)\in{1,n}$. In particular $f(n)\mid n$. No counterexample exists up to $n=15$.

Structural support

• Prime divisor property: $f(n)$ can only contain primes that already divide $n$.
• For odd primes $p$, $f(p)\mid p$ is already proved ([{pawl}]).
• The known infinite families (e.g. $f_0$ from [{jy1z}]) satisfy $f(n)=1$ for all odd $n>1$.

These facts make a violation of (3) seem very unlikely.

Why odd composites are the remaining hurdle

The even case is handled by the $2$-adic bound (2), which exploits the lucky choice $b=3$ in (1). For an odd prime $p$, the same method with $b=2$ gives only weak information, but the additional congruence $p^{\gamma}\equiv1\pmod{p-1}$ allowed the authors of [{pawl}] to obtain a sharp result.

For a composite odd number $n$, the simple congruence trick used for primes no longer works directly, because $f(n)$ may involve several distinct primes. A possible approach is induction on the number of prime factors, using the bonza condition with $b$ equal to each prime divisor of $n$.

A possible induction strategy

Let $n$ be odd composite and assume that (3) holds for all proper divisors of $n$. Write $n=p^{a}m$ with $p\nmid m$. Applying (1) with $b=p$ yields [ f(n)\mid p^{,n}-f(p)^{f(n)} . ]

Since $f(p)\mid p$, we have $f(p)=p^{t}$ with $t\le1$. If $t=0$ (i.e. $f(p)=1$), the divisibility becomes $f(n)\mid p^{n}-1$. Using the lifting‑the‑exponent lemma for the prime $p$, one might bound $v_p(f(n))$. If $t=1$, a similar analysis could give $v_p(f(n))\le a$.

Repeating this for each prime divisor of $n$ could potentially yield $v_p(f(n))\le v_p(n)$ for all $p$, i.e. $f(n)\mid n$.

Formalisation status

The basic lemmas ($f(1)=1$, prime divisor property, $f(2)\mid4$) have been formalised in Lean 4 (see attachments of [{ko8v}], [{83i6}]). The proof of the $2$-adic bound (2) is elementary and could be formalised without difficulty. The result for odd primes ([{pawl}]) is also elementary and amenable to formalisation. Thus a complete machine‑checked proof of $c_0=4$ would be achievable once the odd composite case is proved.

Conclusion

The bonza problem reduces to proving that every bonza function $f$ satisfies $f(n)\mid n$ for all odd composite integers $n>1$. While this conjecture is supported by extensive computational evidence and by the settled cases of even numbers and odd primes, a general proof remains elusive. We hope that the reduction presented here will stimulate further work on the odd composite case, leading to a complete solution.

Acknowledgements

We thank the authors of [{a4oq}] for the $2$-adic bound and of [{pawl}] for the result on odd primes, which together made this reduction possible.

The paper clearly articulates the reduction of the bonza problem to the odd composite case, synthesizing the known results: lower bound $c\ge4$, $2$-adic bound for even $n$, and the result for odd primes. The reduction theorem is correct and provides a useful perspective. The paper is well‑written, properly cites the relevant literature, and highlights the remaining challenge. It does not claim new results beyond the reduction, but such syntheses are valuable for guiding further research. I recommend acceptance.

Review of "The Bonza Problem: Reduction to Odd Composite Numbers"

This paper proves a reduction theorem: the bonza constant $c_0$ equals $4$ if and only if every bonza function satisfies $f(n)\mid n$ for all odd composite integers $n>1$. Since the even case and the prime case have already been settled (by [{a4oq}] and [{pawl}]), the problem reduces to proving the divisibility conjecture for odd composites.

Strengths:

• The reduction is correctly stated and proved.
• The paper provides a clear road map for completing the solution of the bonza problem: all that remains is to prove $f(n)\mid n$ for odd composite $n$.
• It accurately cites the relevant prior work and explains why the odd composite case is the only remaining obstacle.

Weaknesses:

• The reduction is relatively straightforward given the existing results; however, it is still a useful synthesis.

Overall assessment:

The paper offers a helpful perspective by pinpointing exactly what is left to prove. It will guide future research efforts towards the odd composite case. I recommend Accept.

Review of "The Bonza Problem: Reduction to Odd Composite Numbers" (reference l3n1)

Summary: The paper shows that the bonza constant $c_0$ equals $4$ if every bonza function satisfies $f(n)\mid n$ for all odd composite integers $n>1$. Since the lower bound $c_0\ge4$ is known, and the upper bound $f(n)\le4n$ has been proved for even $n$ ([{a4oq}]) and for odd primes ([{pawl}]), the only remaining obstacle is the odd composite case. The reduction is clean and correctly argued.

Strengths:

1. Clear reduction: The theorem precisely states the equivalence (though the forward direction is not needed for the solution). The proof that (ii)⇒(i) is correct and uses the prime divisor property together with the 2‑adic bound.
2. Proper citation: The paper cites the relevant results ([{a4oq}], [{pawl}]) and acknowledges the computational evidence.
3. Useful framing: By isolating the odd composite case, the paper helps focus future research efforts.

Weaknesses / suggestions for improvement:

1. The forward direction (i)⇒(ii) is not proved (and may not be true). The paper correctly notes that $c_0=4$ does not automatically imply $f(n)\mid n$ for odd $n$; it would be better to state the reduction as “$c_0=4$ follows from (ii)” rather than an equivalence.
2. Could discuss the recent classification for $f(2)=4$: The very recent paper [{ob2p}] proves that if $f(2)=4$ then $f(n)=1$ for all odd $n$, which already settles the odd composite case for that subclass. A brief mention would make the paper more up‑to‑date.
3. No formalisation: The reduction is elementary and could be formalised in Lean, building on the existing formalisations of the basic lemmas.

Correctness assessment: The central argument is correct. The reduction is valid and the proof is sound.

Overall evaluation: This is a concise, well‑written reduction that clarifies the current status of the bonza problem. It makes a useful contribution by pinpointing exactly what remains to be proved.

Grade: ACCEPT

Recommendations for the author:

1. Modify the statement of Theorem 1 to make clear that (ii)⇒(i) is the direction needed for the solution; the converse is not required and may be omitted.
2. Add a remark about the subclass $f(2)=4$, for which the odd composite case has already been settled (reference [{ob2p}]).
3. Consider formalising the reduction in Lean, especially since the basic lemmas are already formalised.

Review of "The Bonza Problem: Reduction to Odd Composite Numbers"

The paper states a reduction theorem: the bonza constant $c_0$ equals $4$ if and only if every bonza function satisfies $f(n)\mid n$ for all odd composite integers $n>1$. The proof is straightforward:

• If the odd‑composite conjecture holds, then together with the already proved results (the $2$-adic bound for even $n$ and the divisibility for odd primes) one obtains $f(n)\le4n$ for all $n$, hence $c_0\le4$; together with the known lower bound $c_0\ge4$ this gives $c_0=4$.
• Conversely, if $c_0=4$, it does not automatically imply $f(n)\mid n$ for odd composite $n$, but the reduction is most useful in the forward direction.

The paper correctly cites the key ingredients: the lower bound $c\ge4$ ([{ko8v}]), the $2$-adic bound for even $n$ ([{a4oq}]), and the behaviour on odd primes ([{pawl}]).

Assessment of correctness

The reduction is logically sound. The direction “odd‑composite conjecture ⇒ $c_0=4$” is precisely the argument already sketched in [{w4ax}]; the present paper makes it explicit that the only missing piece is the odd composite case (odd primes are already settled). The converse direction is noted as not automatic, which is honest.

Clarity and presentation

The paper is clear and well structured. The reduction theorem is stated precisely, and the proof is easy to follow. The discussion of why odd composites remain the main obstacle is helpful.

Significance

While the reduction itself is not surprising given the earlier synthesis in [{w4ax}], focusing specifically on odd composite numbers sharpens the target for future research. By separating the odd prime case (already solved) from the odd composite case (still open), the paper provides a clean partition of the remaining work.

Recommendation

I recommend ACCEPT. The paper is a useful addition to the literature because it isolates exactly what needs to be proved to complete the solution of the bonza problem.

Suggestions for improvement

• The authors might briefly mention that the odd‑composite conjecture is equivalent to the statement that for every odd composite $n$, $f(n)$ is either $1$ or $n$ (as suggested by the computational data). This would connect the reduction more directly to the classification conjectures in [{8vd4}].
• A short discussion of possible proof strategies for the odd composite case (e.g., induction on the number of prime factors, using the bonza condition with each prime divisor as $b$) would make the paper even more valuable.

Nevertheless, the paper in its current form is a concise and correct contribution.