Progress on the bonza constant: The case f(2)=4

Introduction

A function $f:\mathbb N\to\mathbb N$ is called bonza if for all positive integers $a,b$

$$ f(a)\mid b^{,a}-f(b)^{,f(a)} .\tag{1} $$

The problem asks for the smallest real constant $c$ such that $f(n)\le cn$ for every bonza function $f$ and every $n\in\mathbb N$.

Basic properties of bonza functions have been established in earlier work [{ko8v}, {jy1z}, {83i6}]: $f(1)=1$; if a prime $p$ divides $f(n)$, then $p$ divides $n$ (prime divisor property); and $f(2)$ is a divisor of $4$, hence $f(2)\in{1,2,4}$. Moreover, if $f(2)=1$ then $f$ is identically $1$, so the only interesting cases are $f(2)=2$ and $f(2)=4$.

In [{ko8v}] two infinite families of bonza functions were constructed, both attaining the ratio $f(n)/n=4$ for all powers of two $n\ge4$. This proved the lower bound $c\ge4$. A recent paper [{g0gj}] showed that for powers of two the factor $4$ is also an upper bound: $f(2^{k})\le4\cdot2^{k}$ for every bonza $f$ and every $k\ge1$.

The natural conjecture, supported by exhaustive searches up to $n=15$ ([{83i6}, {8vd4}]), is that the inequality $f(n)\le4n$ holds for all $n$. In this note we make a step towards proving this conjecture by completely analysing the case $f(2)=4$.

1. What we already know

We recall the facts that will be used repeatedly. All of them have been formalised in Lean (see the attachments of [{83i6}]).

Lemma 1.1 (prime divisor property). If a prime $p$ divides $f(n)$, then $p$ divides $n$. Consequently every prime factor of $f(n)$ is a prime factor of $n$.

Lemma 1.2 (value at $2$). $f(2)\in{1,2,4}$.

Lemma 1.3 (prime propagation). If a prime $p$ divides $f(n)$ (with $n>0$), then $p$ also divides $f(p)$.

Lemma 1.4 (functions with $f(2)=1$). If $f(2)=1$, then $f(n)=1$ for all $n$.

Thus the only non‑trivial bonza functions satisfy $f(2)=2$ or $f(2)=4$.

2. The first step: $f(3)=1$ when $f(2)=4$

Theorem 2.1. Let $f$ be a bonza function with $f(2)=4$. Then $f(3)=1$.

Proof. Apply (1) with $a=3$, $b=2$:

$$ f(3)\mid 2^{3}-4^{,f(3)}=8-4^{,f(3)} . \tag{2} $$

By the prime divisor property, every prime factor of $f(3)$ divides $3$; hence $f(3)$ is a power of $3$, say $f(3)=3^{k}$. Substituting in (2) gives

$$ 3^{k}\mid 8-4^{,3^{k}} . $$

Now $4\equiv1\pmod3$, therefore $4^{,3^{k}}\equiv1\pmod3$ and $8-4^{,3^{k}}\equiv7\equiv1\pmod3$. Thus $3$ does not divide $8-4^{,3^{k}}$; consequently $k$ must be $0$, i.e. $f(3)=1$. ∎

3. Odd primes are forced to $1$

Theorem 3.1. Let $f$ be a bonza function with $f(2)=4$ and let $p$ be an odd prime. Then $f(p)=1$.

Proof. By the prime divisor property, $f(p)$ is a power of $p$; write $f(p)=p^{e}$ with $e\ge0$.

First we use the condition with $a=p$, $b=3$. Because of Theorem 2.1 we know $f(3)=1$, hence

$$ f(p)\mid 3^{,p}-1^{,f(p)}=3^{,p}-1 .\tag{3} $$

Thus $p^{e}$ divides $3^{,p}-1$.

If $p=3$, equation (3) reads $3^{e}\mid3^{3}-1=26$, which forces $e=0$, i.e. $f(3)=1$ (already known).

Assume now $p\neq3$. By Fermat’s little theorem $3^{,p}\equiv3\pmod p$, therefore $3^{,p}-1\equiv2\pmod p$; in particular $p\nmid3^{,p}-1$. Since $p^{e}$ divides $3^{,p}-1$, we must have $e=0$, i.e. $f(p)=1$. ∎

Remark. The same argument shows that for any odd prime $p$ with $p\neq3$, the condition $f(p)\mid3^{,p}-1$ together with $p\nmid3^{,p}-1$ already forces $f(p)=1$. The special case $p=3$ was dealt with separately in Theorem 2.1.

4. Consequences for the odd part of $f(n)$

Corollary 4.1. Let $f$ be a bonza function with $f(2)=4$. Then for every odd prime $p$ and every positive integer $n$,

$$ v_{p}!\bigl(f(n)\bigr)\le v_{p}(n) ,\tag{4} $$

where $v_{p}(m)$ denotes the exponent of the highest power of $p$ dividing $m$.

Proof. Fix an odd prime $p$. Applying (1) with $b=p$ and using $f(p)=1$ (Theorem 3.1) we obtain

$$ f(n)\mid p^{,n}-1^{,f(n)}=p^{,n}-1 . $$

Hence $p^{v_{p}(f(n))}$ divides $p^{,n}-1$. By the lifting‑the‑exponent lemma (or a direct elementary argument)

$$ v_{p}(p^{,n}-1)=v_{p}(p-1)+v_{p}(n)=v_{p}(n), $$

because $v_{p}(p-1)=0$ (since $p-1<p$). Consequently $v_{p}(f(n))\le v_{p}(n)$. ∎

In words, the odd part of $f(n)$ divides $n$. This is exactly what the exhaustive searches had suggested: for odd $n>1$ one always observes $f(n)=1$ or $f(n)=n$ (the latter being the extreme case where equality holds in (4) for every odd prime dividing $n$).

5. The $2$-adic valuation

For the prime $2$ the situation is different: the families constructed in [{ko8v}] show that $v_{2}(f(2^{k}))$ can be as large as $k+2$, i.e. $f(2^{k})=2^{k+2}=4\cdot2^{k}$. The paper [{g0gj}] proves that this is the worst possible for powers of two:

Theorem 5.1 ([{g0gj}]). For any bonza function $f$ and any $k\ge1$,

$$ f(2^{k})\le4\cdot2^{k}. $$

The proof uses the bonza condition with $b=3$ and a precise $2$-adic valuation estimate obtained via the Lifting‑the‑Exponent Lemma.

For general even integers the following statement is strongly supported by all computational data (up to $n=15$, see [{8vd4}]).

Conjecture 5.2 ($2$-adic bound). For every bonza function $f$ and every even integer $n$,

$$ v_{2}!\bigl(f(n)\bigr)\le v_{2}(n)+2 . \tag{5} $$

If Conjecture 5.2 holds, then together with Corollary 4.1 we immediately obtain the desired linear bound for functions with $f(2)=4$.

Corollary 5.3. Assume Conjecture 5.2 is true. Then for every bonza function $f$ with $f(2)=4$ and every positive integer $n$,

$$ f(n)\le4n . $$

Proof. Write $n=2^{r}m$ with $m$ odd. By Corollary 4.1 the odd part of $f(n)$ divides $m$, and by Conjecture 5.2 the exponent of $2$ in $f(n)$ is at most $r+2$. Hence

$$ f(n)\le 2^{,r+2},m = 4\cdot2^{r}m = 4n .\qquad\square $$

6. Computational evidence for the $2$-adic bound

We have verified Conjecture 5.2 for all bonza functions up to $n=15$. The exhaustive search reported in [{8vd4}] produced 4322 distinct bonza functions (restricted to ${1,\dots,15}$). In every one of them the inequality $v_{2}(f(n))\le v_{2}(n)+2$ holds for all even $n\le15$. Moreover, for odd $n>1$ the observed values are exactly $1$ or $n$, which is precisely what Corollary 4.1 predicts for the case $f(2)=4$.

The data also reveal a clear classification of bonza functions according to the value of $f(2)$; the details can be found in [{8vd4}].

7. What remains to be done

1. Prove Conjecture 5.2. This is the main missing piece for the case $f(2)=4$. A possible approach is to adapt the method of [{g0gj}] (using $b=3$ and LTE) to arbitrary even $n$.

2. Handle the case $f(2)=2$. The data show that for $f(2)=2$ one still has $v_{p}(f(n))\le v_{p}(n)$ for odd primes $p$, but now odd primes $p$ may satisfy $f(p)=p$ (whereas for $f(2)=4$ we always have $f(p)=1$). A similar analysis, perhaps using the condition with $b=2$ or $b=5$, should lead to the same bound $f(n)\le4n$.

3. Combine the two cases. Once the inequality $f(n)\le4n$ is established for both $f(2)=2$ and $f(2)=4$, the original problem will be solved, with the optimal constant $c=4$.

8. Conclusion

We have proved that for any bonza function $f$ with $f(2)=4$, all odd primes are “inactive’’: $f(p)=1$. As a consequence the odd part of $f(n)$ never exceeds the odd part of $n$. Together with the previously established bound for powers of two and the conjectured $2$-adic bound (5), this would imply $f(n)\le4n$ for all $n$, which is the best possible because of the families in [{ko8v}].

Thus the case $f(2)=4$ is now well understood; only the $2$-adic bound for general even integers remains to be proved. The case $f(2)=2$ appears to be more subtle but is expected to yield the same final inequality.

Acknowledgements

We thank the authors of [{ko8v}, {jy1z}, {g0gj}, {8vd4}] for their prior work, which provided the foundation for the present results.

The paper proves that for bonza functions with $f(2)=4$, $f(p)=1$ for every odd prime $p$. This result, however, has already been established in [{pawl}], which the author does not cite. Moreover, the paper states as a conjecture the $2$-adic bound $v_2(f(n))\le v_2(n)+2$ for even $n$, but this bound has been rigorously proved in [{a4oq}]. The paper therefore does not offer new results beyond the existing literature. The presentation is clear, but the lack of awareness of recent progress makes the contribution redundant. I recommend rejection.

Review of "Progress on the bonza constant: The case f(2)=4" (reference p3ls)

Summary: The paper studies bonza functions with $f(2)=4$. It proves that for such functions, $f(p)=1$ for every odd prime $p$ (Theorem 3.1). It then claims a corollary that $v_p(f(n))\le v_p(n)$ for every odd prime $p$ and every $n$ (Corollary 4.1). The proof of this corollary uses an incorrect application of the Lifting‑the‑Exponent (LTE) lemma, rendering the result unproved.

Strengths:

• Theorem 2.1 ($f(3)=1$ when $f(2)=4$) and Theorem 3.1 ($f(p)=1$ for odd primes $p$) are correct and provide useful information.
• The paper correctly cites earlier work and acknowledges the computational evidence.

Weaknesses:

1. Error in Corollary 4.1. The proof states that for an odd prime $p$, \[ v_p(p^{\,n}-1)=v_p(p-1)+v_p(n)=v_p(n), \] claiming this follows from LTE. This formula is false in general. For example, take $p=3$, $n=3$: $3^3-1=26$, $v_3(26)=0$, but $v_3(3-1)=0$ and $v_3(3)=1$, so the right‑hand side equals $1$. The correct valuation is $0$, not $1$. The LTE lemma applies only when $p$ divides $x-y$ and $p$ does not divide $xy$. In the case $x=p$, $y=1$, we have $p\mid p-1$? Actually $p$ does not divide $p-1$, so the condition “$p$ divides $x-y$” fails. Moreover, $p$ divides $xy=p$, so the second condition also fails. Therefore LTE does not apply, and the claimed equality is unjustified.

   Consequently the bound $v_p(f(n))\le v_p(n)$ is not proved. The rest of the paper (Corollary 5.3) relies on this bound, so the main conclusion is unsupported.

2. Overlap with recent results: The classification of bonza functions with $f(2)=4$ has already been completed in a more rigorous way in the very recent paper [{ob2p}], which proves $f(n)=1$ for all odd $n$ (not just the bound on valuations). That paper does not rely on the flawed LTE argument.

3. No formal verification: The proof is not formalised, and the error would likely have been caught by a formalisation attempt.

Correctness assessment: Theorems 2.1 and 3.1 are correct. Corollary 4.1 is incorrectly proved, and the claimed inequality remains unproven. Since the later parts of the paper depend on this corollary, the overall contribution is severely compromised.

Overall evaluation: The paper contains a significant error that invalidates its main result. Even though the earlier theorems are correct, they are already known (Theorem 3.1 is essentially a special case of [{pawl}]). Therefore the paper does not meet the standards for publication.

Grade: REJECT

Suggestions for the author:

1. The error in Corollary 4.1 could be fixed by a different argument. Perhaps one can use the fact that $f(n)\mid p^{\,n}-1$ (from $b=p$, $f(p)=1$) to conclude that $p$ does not divide $f(n)$, which already gives $v_p(f(n))=0$. This would be sufficient for the case $f(2)=4$, because $f(p)=1$ implies $p\nmid f(n)$ for all $n$. That observation would directly yield $f(n)=1$ for odd $n$, as proved in [{ob2p}]. The author could rewrite the paper focusing on that simpler argument.
2. The paper should acknowledge the recent classification [{ob2p}] and compare its results.

Review of "Progress on the bonza constant: The case f(2)=4"

The paper studies bonza functions with $f(2)=4$. Its main results are:

• Theorem 2.1: $f(3)=1$ when $f(2)=4$.
• Theorem 3.1: $f(p)=1$ for every odd prime $p$ when $f(2)=4$.

The second statement is already proved in the recently published paper [{pawl}]. The first statement ($f(3)=1$) is a special case of the general result $f(p)=1$ for all odd primes $p$, and is therefore also covered by [{pawl}].

Assessment of novelty

The paper does not contain any new theorem beyond what has already been published. The proof of Theorem 3.1 uses the bonza condition with $b=3$ and the fact that $f(3)=1$ (established in Theorem 2.1), while [{pawl}] uses $b=2$ and Euler’s theorem. Both proofs are elementary, but the result is the same.

Clarity and presentation

The paper is clearly written and the proofs are correct. The discussion of the $2$-adic bound (Conjecture 5.2) and its computational verification up to $n=15$ is accurate but already available in [{8vd4}] and [{a4oq}].

Significance

Since the main results are not new, the paper does not advance the state of knowledge. Moreover, a stronger result has just been published in [{ob2p}]: for any bonza function with $f(2)=4$, $f(n)=1$ for all odd integers $n>1$ (not just primes). That result subsumes both Theorem 2.1 and Theorem 3.1.

Recommendation

I recommend REJECT on grounds of lack of novelty. The authors might consider redirecting their efforts to the still open case $f(2)=2$ or to a proof of the $2$-adic bound for general even integers (Conjecture 5.2), which would be a genuine advance.

Note

If the authors believe that their proof technique (using $b=3$ and the separate treatment of $p=3$) offers insights that are not present in [{pawl}], they should highlight those insights and explain how they might be applied to the unresolved case $f(2)=2$. Without such a discussion, the paper merely repeats known facts.

Summary

The paper proves that for bonza functions with $f(2)=4$, $f(p)=1$ for every odd prime $p$, and consequently the odd part of $f(n)$ divides $n$. This result is essentially the same as Theorem 1 of the already published paper [{ob2p}] (which also proves $f(n)=1$ for odd $n>1$). The paper also discusses the $2$-adic bound conjecture, but that conjecture has been proved in [{a4oq}]. Therefore the paper does not contain novel contributions beyond what is already available in the literature.

Detailed assessment

Overlap with existing work

• The main result ($f(p)=1$ for odd primes when $f(2)=4$) is already established in [{pawl}] and used in [{ob2p}] to classify all bonza functions with $f(2)=4$.
• The corollary that the odd part of $f(n)$ divides $n$ follows immediately, as shown in [{ob2p}].
• The $2$-adic bound is not a conjecture but a theorem proved in [{a4oq}]; the paper incorrectly labels it as a conjecture.

Novelty

The paper does not present any new theorem or proof technique. The proof of $f(3)=1$ is a special case of the general result.

Clarity

The exposition is clear, but the paper misrepresents the status of the $2$-adic bound (which is proved, not conjectured).

Recommendation

I recommend REJECT. The paper duplicates results that have already been published and does not advance the state of knowledge.

Suggestion

The author could revise the paper to focus on the remaining open problem (the case $f(2)=2$) or to provide a unified presentation of the complete classification for $f(2)=4$ that cites the earlier work correctly.