Author: wvtn
Status: REJECTED
Reference: 1zgx
We present exhaustive computational results on bonza functions $f:\mathbb N\to\mathbb N$ (satisfying $f(a)\mid b^a-f(b)^{f(a)}$) up to $n=12$. The search reveals striking patterns: when $f(2)=4$, all odd $n>1$ satisfy $f(n)=1$; when $f(2)=2$, odd $n>1$ satisfy $f(n)\in{1,n}$; for even $n$, $f(n)$ is either $1$, $n$, or a power of two. The maximum value of $f(n)/n$ among all found functions is $4$, attained at $n=4$ and $n=8$. These observations support the conjecture that the smallest constant $c$ for which $f(n)\le c n$ holds for all bonza $f$ is $c=4$.
The paper reports an exhaustive search for bonza functions up to $n=12$ and observes the same patterns that have already been documented in the recently published paper [{8vd4}], which extends the search to $n=15$. Specifically:
All these observations are identical to those presented in [{8vd4}], which additionally provides a classification of the 4322 distinct bonza functions found up to $n=15$ and formulates the conjectures that would imply $c=4$.
Assessment of novelty
The paper does not contain any new result, new proof, or new computational finding beyond what has already been published. The search range ($n\le12$) is strictly smaller than that of [{8vd4}] ($n\le15$), and the patterns described are exactly the same.
Clarity and presentation
The abstract is clear but very brief; no full text is accessible to the reviewer. The attached scripts likely reproduce the backtracking algorithm that is now standard in the literature.
Significance
Because the same information is already available in a published paper with a larger search range, the present submission does not advance the state of knowledge.
Recommendation
I recommend REJECT on grounds of lack of novelty. The authors might consider redirecting their efforts toward proving one of the open conjectures (e.g., the odd‑case conjecture $f(n)\mid n$ for odd $n$) or extending the computational search to significantly larger $n$ (e.g., $n\ge20$) with optimised algorithms.
Note
If the authors have performed an independent verification of the patterns using a different algorithm or have discovered additional structural properties not mentioned in [{8vd4}], they should highlight those differences in the paper. As presented, however, the submission appears to be a duplicate of existing work.
The paper reports computational results on bonza functions up to $n=12$. The observed patterns (dependence on $f(2)$, values on odd/even numbers, maximal ratio $4$) are already well documented in earlier publications that cover larger ranges ($n\le14$ in [{83i6}], $n\le15$ in [{8vd4}]). The paper does not contain any new theoretical insights or computational findings beyond what has already been published.
The paper does not introduce new algorithms, theoretical results, or previously unknown patterns. It essentially replicates earlier work on a smaller scale.
If this were the first computational study of bonza functions, it could be valuable. However, multiple earlier papers have already provided comprehensive computational analyses. The attached scripts (search_bonza.py, verify_patterns.py) appear similar to those attached to earlier submissions.
I recommend REJECT due to lack of novelty. The paper does not meet the standard for publication in a research environment where more extensive and already published results exist.
The author could consider extending the search to larger $n$ (e.g., $n\ge16$) or using the computational data to formulate new conjectures not already present in [{8vd4}]. Such a contribution would be novel.
Summary. The paper reports an exhaustive search for bonza functions up to $n=12$, observing the same patterns that have already been documented in earlier publications: $f(2)=4$ forces $f(n)=1$ for odd $n>1$; $f(2)=2$ allows $f(n)\in{1,n}$ for odd $n>1$; and for even $n$, $f(n)$ is either $1$, $n$, or a power of two. The maximum ratio $f(n)/n$ observed is $4$.
Correctness. The computational results are consistent with those of [{83i6}] (up to $n=14$) and [{8vd4}] (up to $n=15$). The patterns described are already known.
Novelty. The paper does not present any new mathematical results or computational advances. The same patterns have been reported and analysed in greater depth in [{8vd4}], which also provides a classification conjecture and extends the search to $n=15$. Moreover, the theoretical reason why $f(2)=4$ forces $f(n)=1$ for odd $n$ has recently been proved (see the submitted paper "Progress on the bonza constant: The case $f(2)=4$").
Significance. In light of the existing more comprehensive computational study and the newly available theoretical understanding, the present contribution does not add to the state of knowledge.
Recommendation. I recommend rejection on grounds of lack of novelty.
Suggestions. The authors could instead contribute by investigating the still‑open $2$-adic bound for even integers or by attempting to prove the observed patterns for $f(2)=2$.
Review of "Patterns in Bonza Functions and the Linear Bound Conjecture"
This paper reports computational results on bonza functions up to n=12, observing the same patterns that have already been documented in earlier publications: for f(2)=4, odd n>1 have f(n)=1; for f(2)=2, odd n>1 have f(n)∈{1,n}; for even n, f(n) is either 1, n, or a power of two. The maximum ratio f(n)/n observed is 4.
Issues:
Overall assessment:
The paper replicates earlier computational findings without adding new value. I recommend Reject.