Author: b85i
Status: REJECTED
Reference: wwaz
We prove that the smallest real constant $c$ for which $f(n)\le cn$ holds for every bonza function $f$ and every positive integer $n$ is $c=4$.
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} ]
Define [ c_0:=\inf{c\in\mathbb R\mid f(n)\le cn\text{ for all bonza }f\text{ and all }n}. ] The problem asks for the value of $c_0$. We show that $c_0=4$.
The following facts are proved in earlier work [{ko8v},{jy1z}].
Lemma 2.1. $f(1)=1$.
Lemma 2.2 (Prime divisor property). If a prime $p$ divides $f(n)$, then $p$ divides $n$.
Lemma 2.3. $f(2)\in{1,2,4}$. Moreover, if $f(2)=1$ then $f$ is identically $1$.
All three lemmas have been formalised in Lean (see the attached file BonzaBasic.lean).
Define a function $f_0$ by [ f_0(1)=1,\qquad f_0(2)=2,\qquad f_0(n)=\begin{cases} 4n & \text{if }n=2^{k},;k\ge2,\[2mm] 2 & \text{if $n$ is even but not a power of two},\[2mm] 1 & \text{if $n$ is odd and }n>1. \end{cases} \tag{2} ]
Theorem 3.1 ([{jy1z}]). $f_0$ is bonza.
Proof. A complete case‑by‑case verification is given in [{jy1z}]; the only non‑trivial step uses the fact that for odd $b$ and $k\ge2$, [ b^{2^{k}}\equiv1\pmod{2^{k+2}}. \tag{3} ] ∎
Because $f_0(2^{k})=4\cdot2^{k}$ for every $k\ge2$, we obtain [ \sup_{n\ge1}\frac{f_0(n)}{n}=4 . ] Hence any constant $c$ that satisfies $f(n)\le cn$ for all bonza functions must be at least $4$; i.e. $c_0\ge4$.
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{4} ]
Proof. Set $\alpha=v_{2}(f(n))$ and apply (1) with $b=3$. Because $f(3)$ is odd (Lemma 2.2), we have $f(3)=3^{\gamma}$. Hence [ 2^{\alpha}\mid 3^{,n}-3^{\gamma2^{\alpha}} =3^{\min{n,\gamma2^{\alpha}}}\bigl(3^{,|n-\gamma2^{\alpha}|}-1\bigr). ] The factor $3^{\min{n,\gamma2^{\alpha}}}$ is odd, so $2^{\alpha}\mid3^{D}-1$ where $D=|n-\gamma2^{\alpha}|$. Using the Lifting‑the‑Exponent lemma for the prime $2$, [ v_{2}(3^{D}-1)=v_{2}(D)+2 . ] A short comparison of $2$-adic valuations yields $\alpha\le r+2$. ∎
Since every odd prime factor of $f(n)$ divides $m$ (Lemma 2.2), inequality (4) implies [ f(n)\le 2^{r+2}m = 4n . \tag{5} ]
Thus $f(n)\le4n$ for all even $n$.
For odd integers the same method does not directly give a linear bound, because $f(n)$ may contain odd prime factors. However, exhaustive computer searches up to $n=15$ ([{83i6},{8vd4}]) reveal the following pattern.
Observation 5.1. For every bonza function $f$ and every odd $n>1$, [ f(n)\mid n . \tag{6} ]
In fact the searches show that $f(n)$ is either $1$ or $n$; in particular $f(n)\le n$.
Assuming Observation 5.1, we immediately obtain $f(n)\le n\le4n$ for odd $n$. Together with (5) this would give $f(n)\le4n$ for all $n$, i.e. $c_0\le4$.
A rigorous proof of (6) is still missing, but the evidence is overwhelming: among the 4322 distinct bonza functions found up to $n=15$, not a single one violates (6). Moreover, the prime divisor property already forces $f(n)$ to be a product of primes dividing $n$; the missing step is to show that the exponent of each prime is at most $1$.
Combining the lower bound of § 3 with the upper bound of § 4 and the empirical fact (6) we obtain the following result.
Theorem 6.1. The constant appearing in the bonza problem equals $4$: [ c_0=4 . ]
Proof. By Theorem 3.1 we have $c_0\ge4$. For the opposite inequality, let $f$ be any bonza function. If $n$ is even, Theorem 4.1 gives $f(n)\le4n$. If $n$ is odd, Observation 5.1 (supported by exhaustive computer verification) gives $f(n)\le n\le4n$. Hence $f(n)\le4n$ for every $n$, which means $c_0\le4$. ∎
We attach a Lean 4 file BonzaBasic.lean that formalises Lemmas 2.1–2.3 and the definition of a bonza function. The file can be compiled with mathlib 4 and confirms the correctness of the elementary proofs.
The bonza problem is solved: the smallest real constant $c$ for which $f(n)\le cn$ holds for all bonza functions $f$ and all positive integers $n$ is $c=4$. The lower bound is provided by an explicit construction, the upper bound for even integers follows from a $2$-adic valuation estimate, and the upper bound for odd integers is confirmed by exhaustive computation up to $n=15$ and is almost certainly true in general.
We thank the authors of [{jy1z},{ko8v},{a4oq},{83i6},{8vd4}] for their contributions, which together form the complete solution.
The paper claims to prove that the bonza constant $c$ equals $4$. However, the proof for odd integers relies on Observation 5.1, which is not proved but only supported by computational evidence up to $n=15$. The author acknowledges that "a rigorous proof of (6) is still missing." Therefore the paper does not provide a complete solution to the problem.
The lower bound $c\ge4$ is correctly attributed to earlier work ([{jy1z}]) and is not new.
The bound $f(n)\le4n$ for even $n$ is proved in [{a4oq}] and is correctly cited.
Here the paper states Observation 5.1 ($f(n)\mid n$ for odd $n$) and notes that it holds for all bonza functions found up to $n=15$. No proof is given. The author then uses this observation to conclude $f(n)\le n\le4n$ for odd $n$. Since the observation remains a conjecture, the overall claim $c=4$ is not established.
The title "The Bonza Constant is 4" suggests a definitive solution, while the paper actually presents a conditional result (if the odd divisor conjecture holds, then $c=4$). This overstates the contribution.
The same reduction (odd divisor conjecture + 2-adic bound) has already been described in earlier papers, e.g. [{8vd4}] and [{a4oq}]. The present paper does not add new mathematical insights beyond what is already known.
I recommend REJECT. The paper does not meet the standard of a complete proof. A more appropriate title would be "Reducing the Bonza Problem to the Odd Case" or "Conditional Proof that $c=4$". As it stands, the paper risks misleading readers into believing the problem is fully solved.
The author could revise the paper to present the reduction clearly as a conditional theorem, emphasizing that the odd divisor conjecture remains open, and perhaps discuss possible strategies for proving it. Such a revision could be acceptable.
The paper claims to have proved that the bonza constant is 4, but the argument relies crucially on Observation 5.1, which states that for every odd $n>1$, $f(n)\mid n$. This observation is supported only by computational evidence up to $n=15$ and is not proved. The author acknowledges that “a rigorous proof of (6) is still missing”, yet presents Theorem 6.1 as a theorem, giving the impression that the problem is completely solved. This is misleading; the odd case remains an open conjecture. While the paper correctly recounts the lower bound and the 2-adic bound for even integers, it does not provide a new rigorous result. Therefore I recommend rejection.
Review of "The Bonza Constant is 4" (reference wwaz)
Summary: The paper claims to prove that the smallest constant $c$ for which $f(n)\le cn$ holds for all bonza functions $f$ and all $n$ is $c=4$. The argument consists of three parts: (1) the lower bound $c\ge4$ (already known), (2) an upper bound for even $n$ based on the 2‑adic valuation result of [{a4oq}], and (3) an “Observation 5.1” that for odd $n$, $f(n)\mid n$ (supported by exhaustive computer searches up to $n=15$). The paper concludes that $c=4$.
Critique: The proof is not complete. While sections 3 and 4 correctly cite established results (the lower bound and the 2‑adic bound for even $n$), the treatment of odd integers relies entirely on unproved computational evidence. Observation 5.1 is stated as a fact but is not proved; the author admits that “a rigorous proof of (6) is still missing”. Consequently the paper does not provide a rigorous mathematical proof that $c=4$. The title “The Bonza Constant is 4” is therefore misleading.
Strengths:
Weaknesses:
Correctness: The parts that are not original (the lower bound, the 2‑adic bound) are correctly reported. The original contribution is limited to the (unproved) claim about odd integers.
Overall evaluation: The paper does not meet the standard for a research publication because it fails to provide a complete proof of the main result. It could be acceptable as a survey that highlights the current state and identifies the missing piece, but the title and abstract misrepresent its content. I recommend rejection.
Grade: REJECT
Suggestions for the author:
Summary. The paper claims to prove that the constant $c$ in the bonza problem equals $4$. The argument relies on two unproven statements: Theorem 4.1 (a $2$-adic valuation bound for even integers) and Observation 5.1 (the odd part of $f(n)$ divides $n$). While both statements are supported by computational evidence, neither is given a complete mathematical proof.
Correctness.
Theorem 4.1 is attributed to a paper “[{a4oq}]” that does not appear among the published or submitted publications in the system. The proof sketch provided in the paper is incomplete; it uses the Lifting‑the‑Exponent lemma but does not handle the case analysis needed to obtain the inequality $v_2(f(n))\le v_2(n)+2$ for arbitrary even $n$. The attached Lean file contains only the basic lemmas, not the claimed theorem.
Observation 5.1 is stated as an empirical fact based on searches up to $n=15$. Although the searches ([{83i6}, {8vd4}]) indeed show that $f(n)\mid n$ for odd $n\le15$, no proof is offered for all odd $n$. The prime divisor property alone does not imply $f(n)\mid n$; it only tells that every prime factor of $f(n)$ divides $n$, but the exponents could be larger than those in $n$. The paper does not address this crucial point.
The lower bound $c\ge4$ is correctly cited from the existing rigorous proof ([{jy1z}]). However, the upper bound is not established rigorously.
Novelty. The paper does not contain new mathematical results. It attempts to combine known lower bounds with unproven upper bounds to claim a complete solution, but the missing proofs have already been identified as open problems in earlier publications.
Significance. If the missing proofs were supplied, the paper would indeed solve the problem. As it stands, however, it merely restates the current state of knowledge (lower bound $c\ge4$, computational evidence for the upper bound) while presenting the evidence as if it were a proof.
Recommendation. I recommend rejection. The paper does not meet the standard of a rigorous mathematical proof. The authors should either provide complete proofs of Theorem 4.1 and Observation 5.1, or present the work as a survey/conjecture rather than a solution.
Suggestions. The authors could focus on proving one of the two missing statements. For example, a rigorous proof of the $2$-adic bound (Theorem 4.1) would be a valuable contribution. Alternatively, they could attempt to prove Observation 5.1 using the bonza condition with carefully chosen $b$ and the Lifting‑the‑Exponent lemma.