PREreview of Advancements in Prime Number Study and the Non-Existence of Odd Perfect Numbers

Introduction to the Review

This manuscript tackles some of the most enduring and captivating puzzles in number theory: the nonexistence of odd perfect numbers, the structure and distribution of twin primes, and the interplay between prime gaps and the Riemann Hypothesis. These are not isolated curiosities—they represent foundational questions tied deeply to the behavior of prime numbers, the nature of mathematical infinity, and the analytic structure of the integers.

The effort to advance our understanding of these topics through original formulations and structural arguments is both bold and admirable. The author brings to the table fresh approaches and hypotheses, with a clear determination to reason independently and creatively. This kind of mathematical ambition deserves recognition, as it reflects the true spirit of mathematical discovery—posing meaningful questions, constructing new relationships, and exploring boundaries that have resisted resolution for centuries.

In reviewing this manuscript, the goal is not merely to critique but to support the ongoing intellectual journey. Each section of the manuscript contributes in its own way to a greater tapestry of ideas. The suggestions below are offered with the intent of helping the author sharpen the formulations, clarify the logic, and build a more rigorous foundation for the original insights presented. Even when some claims require further refinement, the underlying motivation and structure are of genuine mathematical interest.

Points for Review and Suggested Revisions

1. Recursive Equation for Odd Perfect Numbers

Problem for Review: The paper introduces a novel recursive identity involving the divisors of a hypothetical odd perfect number:   T = 2m + 1, leading to the expression:   q² − 2mq + 2m + 1 + q_L + q_V + q_A + q_B + ... = 0 While this is an inventive formulation, the derivation and interpretation of the additional terms (q_L, q_V, ...) remain ambiguous.

Suggested Revision: Clearly define how these added terms are derived. Connecting the identity more explicitly to Euler’s form or the sigma function σ(n) would enhance rigor.

2. Parity of the Divisor q

Problem for Review: The manuscript argues that q must be even, contradicting the assumption that T is odd. This inference may not generalize without further support.

Suggested Revision: Investigate if this contradiction always holds. Consider whether additional constraints on q must be established. Strengthen the argument with known results.

3. Twin Primes and Digit-End Classification

Problem for Review: The classification of twin primes by ending digits leads to a contradiction in summed parity. The argument, while creative, is not fully rigorous.

Suggested Revision: Clarify if this is heuristic or intended as a formal proof. If the latter, it must include deeper tools like sieve methods or Brun’s theorem.

4. Prime Gap Bound: Pₙ₊₁ − Pₙ < 2Pₙ

Problem for Review: This is based on π(x) and Rosser–Schoenfeld estimates, but applying them strictly to all n might go beyond their guaranteed range.

Suggested Revision: Clarify whether the inequality is empirical or conjectural. Provide range and cite theorems or data to support the claim.

5. Connection to the Riemann Hypothesis via Prime Intervals

Problem for Review: The argument risks circularity since it depends on a non-established gap inequality.

Suggested Revision: Frame conditionally: “If the gap bound holds, then the interval (x − 4x / (π log x), x] contains at least one prime for sufficiently large x.”

Conclusion to the Review

This manuscript represents a passionate and promising engagement with deep mathematical problems. The author demonstrates a strong grasp of number theory and creativity. Even speculative parts offer valuable directions.

Mathematics progresses through refining hypotheses, checking logic, and dialogue. This work fits that mold. It’s a strong, sincere step forward.

To the author: Keep developing, testing, and writing. Your initiative and originality are your greatest strengths. With continued refinement, this work can meaningfully contribute to number theory.

Competing interests

The author declares that they have no competing interests.