PREreview of Self-Healing Numbers: A New Class of Integers with Positional Divisibility Properties

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\title[Pre-review Referee Report] {Pre-review Referee Report: ``Self-Healing Numbers: A New Class of Integers with Positional Divisibility Properties''}

\author{Anik Chakraborty}

\date{\today}

\begin{document}

\maketitle

\noindent \textbf{Manuscript:} Self-Healing Numbers: A New Class of Integers with Positional Divisibility Properties \\

\textbf{Author:} Arya Pathak \\

\textbf{Manuscript ID:} preprints202509.1648.v1 \\

\textbf{Recommendation:} \textbf{Major Revision Required}

\section{Summary}

The manuscript introduces ``self-healing numbers'' (SHNs), a new class of positive integers defined by the property that for each digit position $i$, removing the digit at position $i$ yields a number divisible by $i$. The author provides computational enumeration up to seven digits, proves basic structural properties, demonstrates non-hereditary behavior, and analyzes growth patterns. While the definition is novel and the computational work appears sound, the manuscript contains a critical mathematical error in its main theorem, insufficient theoretical depth, inadequate citation of related literature, and premature statistical claims. The work is suitable for recreational mathematics or as an undergraduate computational project, but requires substantial revision before meeting standards for peer-reviewed publication.

\section{Major Issues Requiring Correction}

\subsection{Critical Error in Theorem 4.1}

\textbf{Issue:} Theorem 4.1 states that ``Every self-healing number with two or more digits must have an even last digit.'' This statement is \emph{mathematically false} as written.

\textbf{Evidence:} Direct computational verification reveals 20 two-digit self-healing numbers with odd last digits:

\[

21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 61, 63, 65, 67, 69, 81, 83, 85, 87, 89.

\]

For example, $n = 21$ satisfies:

\begin{itemize}

\item Position 1: $n(1) = 1$, and $1 \mid 1$ \checkmark

\item Position 2: $n(2) = 2$, and$2 \mid 2$ \checkmark

\end{itemize}

Thus $21$is a self-healing number with odd last digit$1$, directly contradicting Theorem 4.1.</p><p>\textbf{Root Cause:} The proof incorrectly asserts that ``$n(2)$has the same last digit as$n$.'' For$k = 2$(two-digit numbers), if$n = d_1 d_2$, then$n(2)$removes$d_2$, leaving only$d_1$as a single-digit number. The last digit of$n$is$d_2$, while the last digit of$n(2)$is$d_1$--- these are \emph{different} digits. The proof's logic applies only for$k \geq 3$, where removing the second digit preserves the last digit.</p><p>\textbf{Required Action:} The theorem statement must be restricted to$k \geq 3$, and the proof must explicitly address the case$k = 2$ separately (showing both even and odd last digits occur). The reliance of Section 6's ``crucial computational optimization'' on this false theorem also invalidates that optimization for two-digit searches.

\textbf{Severity:} This is the manuscript's only non-trivial structural result. Its incorrectness is a fundamental flaw that must be addressed before publication.

\subsection{Inadequate Theoretical Development}

\textbf{Issue 1 -- Congruence Framework Underutilized:} Proposition 2.4 introduces a system of congruences but stops short of meaningful analysis. The congruence conditions

\[

n(i) \equiv 0 \pmod{i}, \quad i = 1, 2, \ldots, k

\]

form a system amenable to Chinese Remainder Theorem analysis for coprime moduli, yet the paper does not pursue solvability conditions, asymptotic counting formulas, or constraint interdependencies. This represents a significant missed opportunity for theoretical depth.

\textbf{Recommendation:} The author should cite standard references on systems of congruences (e.g., Hardy \& Wright, Chapter 5; Ireland \& Rosen's \emph{A Classical Introduction to Modern Number Theory}; Keith Conrad's exposition on the Chinese Remainder Theorem) and discuss whether CRT provides insight into existence or density of SHNs for arbitrary $k$.</p><p>\textbf{Issue 2 -- Trivial Proofs:} Theorem 2.1 (single-digit integers are SHNs) and Theorem 2.2 (two-digit characterization) are trivial by direct inspection and add no mathematical content. Their proofs consume space without advancing understanding.</p><p>\textbf{Issue 3 -- No Asymptotic Analysis:} The manuscript computes densities$\rho(k)$ empirically but provides no theoretical bounds, recurrence relations, or generating functions. Standard techniques from analytic number theory (sieve methods, probabilistic models with refined independence assumptions) could yield asymptotic formulas or growth bounds.

\subsection{Premature and Unsupported Statistical Claims}

\textbf{Issue:} The manuscript claims (Section 5) that growth ratios ``stabilize around 4.8 for lengths 4--6.'' This conclusion rests on exactly \emph{three data points} ($k = 4, 5, 6$) with ratios$4.83, 4.80, 4.80$, respectively. No statistical analysis (confidence intervals, variance, significance testing) supports the claim of ``stabilization.''</p><p>\textbf{Standard Practice:} Computational number theory typically requires 10--20 terms to claim pattern convergence, accompanied by error analysis or heuristic models explaining the observed behavior.</p><p>\textbf{Recommendation:} Either compute more terms ($k \geq 10$) with rigorous statistical analysis, or soften the language to ``ratios \emph{appear} to approach approximately 4.8 based on limited data.'' Acknowledge that three points are insufficient for definitive conclusions.

\subsection{Insufficient Literature Review and Citation}

The manuscript cites only three references (Hardy \& Wright, Guy, Sloane's OEIS), which is grossly inadequate for a research paper. Critical omissions include:

\begin{enumerate}[label=(\alph*)]

\item \textbf{Most Similar Sequence:} OEIS A061651 defines numbers where each digit $d$ divides the number obtained by removing that digit. This is the closest analog (digit-value based vs.\ position-based), yet it is not mentioned. The author must cite and compare with A061651.

\item \textbf{Polydivisible Numbers:} While mentioned, polydivisibles (OEIS A144688) lack proper citations. The hereditary property (every prefix is polydivisible) should be contrasted with SHNs' non-hereditary behavior using established literature (e.g., Wikipedia entry, recreational math texts).

\item \textbf{Harshad Numbers:} The manuscript mentions Harshad numbers but omits foundational work proving their infinitude (e.g., Grundman 1994; Cooper \& Kennedy 1993 on consecutive Niven numbers). These methods are directly relevant to Conjecture 5.1.

\item \textbf{Computational Number Theory:} The complexity analysis (Section 6) lacks citations for algorithmic number theory (e.g., Crandall \& Pomerance's \emph{Prime Numbers: A Computational Perspective}; Shoup's \emph{A Computational Introduction to Number Theory and Algebra}).

\item \textbf{Sieve Theory:} The ``naive probabilistic model'' and constraint dependencies require citation of sieve methods (e.g., Halberstam \& Richert; Friedlander \& Iwaniec's asymptotic sieve work; Tenenbaum's \emph{Introduction to Analytic and Probabilistic Number Theory}).

\item \textbf{Divisibility Sequences:} The broader context of divisibility sequences (Ward 1937, 1939 on linear divisibility sequences; Everest et al.\ on recurrence sequences) should frame SHNs within established theory.

\end{enumerate}

\textbf{Required Action:} Expand references to at least 15--20 citations covering related sequences, computational methods, asymptotic analysis, and divisibility theory. See detailed citation recommendations in Section 4.

\subsection{Conjecture 5.1 Lacks Justification}

The conjecture that SHNs are infinite is stated based solely on ``empirical patterns'' through $k = 7$. No theoretical argument addresses whether constraint growth (satisfying$k$divisibility conditions) eventually exhausts the solution space ($9 \times 10^{k-1}$ candidates). The paper should:

\begin{itemize}

\item Cite methods used for proving infinitude of similar sequences (e.g., Harshad numbers, whose infinitude is proven constructively).

\item Discuss whether density decay ($\rho(k) \to 0$as$k \to \infty$) precludes infinitude or merely reflects thinning.

\item Acknowledge that computational evidence through $k = 7$ provides weak support for infinitude claims.

\end{itemize}

\section{Minor Issues}

\subsection{Terminology and Exposition}

\begin{itemize}

\item The term ``self-healing'' is evocative but lacks mathematical motivation. A brief explanation of why this metaphor is appropriate would improve readability.

\item Definition 1.1's notation ``$n(i)$'' is clear, but the manuscript inconsistently treats$n(i) = 0$ (e.g., Example 1.2). Clarify whether leading zeros are acceptable or if such cases are boundary conditions.

\end{itemize}

\subsection{Computational Details}

\begin{itemize}

\item No algorithmic implementation (pseudocode or actual code) is provided. Best practices in computational mathematics require reproducibility via code sharing (e.g., GitHub, arXiv ancillary files).

\item The $O(k^2 \cdot 10^k)$complexity claim (Section 6) is correct but discusses only one optimization (even last digit, which is incorrect for$k = 2$). Modular arithmetic and constraint propagation could reduce search spaces significantly.

\end{itemize}

\subsection{Table and Data Presentation}

\begin{itemize}

\item Table 2 (growth ratios) includes $k = 1$with ratio$4.44$, but this transition from single-digit to two-digit numbers is fundamentally different from higher transitions (Theorem 2.2 shows exact structure for$k = 2$). This ratio should be separated or explained distinctly.

\item The manuscript lists the first 20 terms of the sequence but does not deposit the full enumeration in a data repository or the OEIS. Submission to OEIS is strongly recommended post-revision.

\end{itemize}

\subsection{Section 8: Open Problems}

The open problems are appropriate but lack context within the broader unsolved problems literature. Citing Guy's \emph{Unsolved Problems in Number Theory} (which the author already references) in this section would better frame these questions.

\section{Detailed Citation and Reference Recommendations}

The following references should be added at specified locations to strengthen the manuscript:

\subsection{Section 1: Introduction}

\textbf{After ``Classical examples include...'':}

\begin{enumerate}

\item Hardy, G.~H., \& Wright, E.~M.\ (2008). \emph{An Introduction to the Theory of Numbers} (6th ed.). Oxford University Press. [Already cited; ensure used for divisibility theory]

\item Sloane, N.~J.~A.\ (2023). \emph{The On-Line Encyclopedia of Integer Sequences}, \url{https://oeis.org}. Specifically cite OEIS A144688 (polydivisible numbers), A005349 (Harshad numbers), and A061651 (digit-removal divisibility).

\item Parker, M.\ (2014). \emph{Things to Make and Do in the Fourth Dimension}. Particular Books. [Accessible treatment of polydivisibles]

\end{enumerate}

\textbf{After Harshad numbers mention:}

\begin{enumerate}[resume]

\item Cooper, C.~N., \& Kennedy, R.~E.\ (1993). On Consecutive Niven Numbers. \emph{Fibonacci Quarterly}, 21, 146--151.

\item Grundman, H.~G.\ (1994). Sequences of Consecutive $n$-Niven Numbers. \emph{Fibonacci Quarterly}, 32, 174--175. [Methods for proving infinitude]

\end{enumerate}

\subsection{Section 2: Proposition 2.4}

\textbf{After introducing congruence system:}

\begin{enumerate}[resume]

\item Conrad, K.\ \emph{Chinese Remainder Theorem}. \url{https://kconrad.math.uconn.edu/blurbs/ugradnumthy/crt.pdf}.

\item Ireland, K., \& Rosen, M.\ (1990). \emph{A Classical Introduction to Modern Number Theory} (2nd ed.), Chapter 3. Springer-Verlag.

\end{enumerate}

\subsection{Section 4: Theorem 4.2 (Non-hereditary Property)}

\textbf{After proof of Theorem 4.2:}

\begin{enumerate}[resume]

\item Sloane, N.~J.~A.\ OEIS A144688 (Polydivisible numbers). Note explicitly the hereditary property for contrast.

\item \url{https://en.wikipedia.org/wiki/Polydivisible_number} [If appropriate for publication venue]

\end{enumerate}

\subsection{Section 5: Growth Analysis}

\textbf{After Table 2:}

\begin{enumerate}[resume]

\item Crandall, R., \& Pomerance, C.\ (2005). \emph{Prime Numbers: A Computational Perspective} (2nd ed.), Chapters 1--2. Springer. [Asymptotic analysis methods]

\item Friedlander, J., \& Iwaniec, H.\ (1998). Asymptotic Sieve for Primes. \emph{Annals of Mathematics}, 148(3), 1041--1065.

\item Tenenbaum, G.\ (1995). \emph{Introduction to Analytic and Probabilistic Number Theory}. Cambridge University Press.

\end{enumerate}

\textbf{After Conjecture 5.1:}

\begin{enumerate}[resume]

\item Guy, R.~K.\ (2004). \emph{Unsolved Problems in Number Theory} (3rd ed.), Section A. Springer-Verlag. [Already cited; use here to frame conjecture]

\end{enumerate}

\textbf{After probabilistic model discussion:}

\begin{enumerate}[resume]

\item Halberstam, H., \& Richert, H.-E.\ (1974). \emph{Sieve Methods}. Academic Press.

\item Motohashi, Y.\ (1983). \emph{Lectures on Sieve Methods and Prime Number Theory}. Tata Institute.

\end{enumerate}

\subsection{Section 6: Algorithmic Considerations}

\textbf{After complexity claim:}

\begin{enumerate}[resume]

\item Shoup, V.\ (2008). \emph{A Computational Introduction to Number Theory and Algebra} (2nd ed.), Chapter 1. Cambridge University Press.

\item von zur Gathen, J., \& Gerhard, J.\ (2013). \emph{Modern Computer Algebra} (3rd ed.). Cambridge University Press.

\end{enumerate}

\subsection{Section 7: Related Work (Requires Major Expansion)}

\textbf{Throughout section:}

\begin{enumerate}[resume]

\item \textbf{CRITICAL:} Cite OEIS A061651 and provide detailed comparison with SHNs.

\item Kennedy, R.~E., Goodman, R., \& Best, C.\ (1980). Mathematical Discovery and Niven Numbers. \emph{MATYC Journal}, 14, 21--25.

\item Vardi, I.\ (1991). Niven Numbers. \emph{Computational Recreations in Mathematica}, pp.~19, 28--31. Addison-Wesley.

\end{enumerate}

\subsection{New Section Recommended: Connection to Divisibility Sequences}

\textbf{Add after Section 7 or expand Related Work:}

\begin{enumerate}[resume]

\item Ward, M.\ (1937). Linear Divisibility Sequences. \emph{Transactions of the AMS}, 41, 276--286.

\item Ward, M.\ (1939). A Note on Divisibility Sequences. \emph{Bulletin of the AMS}, 45, 334--336.

\item Everest, G., et al.\ (2003). \emph{Recurrence Sequences}. American Mathematical Society.

\end{enumerate}

\subsection{Post-Publication Recommendation}

\begin{enumerate}[resume]

\item Submit the sequence to OEIS following guidelines at \url{https://oeis.org/wiki/Overview_of_the_contribution_process}. Include: definition, first 40+ terms, computational code, and link to published paper.

\end{enumerate}

\section{Assessment of Novelty}

The definition of self-healing numbers appears to be novel. Searches of the OEIS, arXiv, and recreational mathematics literature reveal no prior work on this exact construction. However, the novelty is \emph{incremental} rather than foundational:

\begin{itemize}

\item OEIS A061651 (numbers where digit $d$divides the number with$d$ removed) is structurally very similar, differing only in using digit-value rather than position.

\item Polydivisible numbers (OEIS A144688) use positional divisibility via prefixes rather than removal.

\item The definition combines features of existing sequences without deeper connections to number-theoretic structures (modular forms, $L$-functions, Diophantine equations, etc.).

\end{itemize}

The work is appropriate for recreational mathematics journals (e.g., \emph{Mathematics Magazine}, \emph{College Mathematics Journal}, \emph{Journal of Recreational Mathematics}) or computational number theory venues after substantial revision. It is currently unsuitable for pure mathematics journals focused on theoretical advances.

\section{Recommendation and Required Revisions}

I recommend \textbf{Major Revision} with the following mandatory changes:

\subsection{Critical (Must Address Before Acceptance)}

\begin{enumerate}

\item \textbf{Correct Theorem 4.1:} Restrict statement to $k \geq 3$and revise proof to handle$k = 2$ separately. Adjust Section 6's reliance on this result.

\item \textbf{Expand References:} Add at least 15--20 citations covering related sequences (especially A061651), computational methods, asymptotic analysis, and divisibility theory. Use specific recommendations from Section 4.

\item \textbf{Address Statistical Claims:} Either compute more terms with rigorous analysis or soften claims about ``stabilization'' of growth ratios.

\item \textbf{Expand Section 7 (Related Work):} Provide detailed comparison with OEIS A061651, A144688, A005349, including structural differences, growth rates, and hereditary properties.

\end{enumerate}

\subsection{Strongly Recommended}

\begin{enumerate}[resume]

\item Develop the congruence framework (Proposition 2.4) beyond mere statement. Discuss solvability, constraint dependencies, or connections to CRT.

\item Provide computational code for reproducibility (as supplementary material or repository link).

\item Add a new section connecting SHNs to the broader theory of divisibility sequences (Ward et al.).

\item Strengthen Conjecture 5.1 with theoretical discussion (not just computational observation).

\end{enumerate}

\subsection{Optional Improvements}

\begin{enumerate}[resume]

\item Remove or consolidate trivial results (Theorems 2.1, 2.2) to save space for deeper content.

\item Explore base-$b$ generalizations computationally (currently listed only as ``open problem'').

\item Discuss potential applications or connections to other areas (cryptography, coding theory, etc.), if any exist.

\end{enumerate}

\section{Conclusion}

The manuscript introduces an interesting combinatorial object with clear computational evidence of regular structure. However, it suffers from a critical mathematical error (Theorem 4.1), insufficient theoretical depth, inadequate literature engagement, and premature statistical conclusions. The definition's novelty is moderate, situating the work within recreational/computational number theory rather than pure mathematics.

With careful revision addressing the critical issues---particularly correcting Theorem 4.1, expanding citations by at least 15 references, and providing rigorous comparison with OEIS A061651---this manuscript could become a solid contribution to computational number theory suitable for venues like \emph{Mathematics Magazine}, \emph{Integers}, or the \emph{Fibonacci Quarterly}. In its current form, it does not meet publication standards for peer-reviewed journals.

I encourage the author to undertake these revisions, as the core idea has merit and the computational work appears sound. The sequence should also be submitted to the OEIS post-publication to ensure discoverability and community engagement.

\vspace{1em}

\noindent \textit{Reviewer's Note:} This review is provided in a spirit of constructive criticism aimed at strengthening the manuscript. The author has made a genuine effort to introduce and study a new mathematical object, and with appropriate revisions, this work can make a valuable contribution to the literature on integer sequences with special divisibility properties.

\vspace{1cm}

\noindent

\textbf{Pre-Reviewer:} Anik Chakraborty,\\

\textbf{Affiliation:} Researcher, Department of Mathematics, University of Delhi, India\\

Date: October 21, 2025

\end{document}

Competing interests

The author declares that they have no competing interests.

Use of Artificial Intelligence (AI)

The author declares that they did not use generative AI to come up with new ideas for their review.