Redundancy Degree and Reconstructibility in k-Exact-Perfect Numbers

Overview

Purpose: The knowledge of the relationship between an integer n and its divisors aids number-theoretic research in integer partitions, prime number search, and subset sum problems. Prior work defines n as an abundant integer if its abundance, the difference between the sum of its divisors and 2n, is positive. Abundant integer n is k-exact-perfect number (k-epn) if it possesses a redundant set of k distinct divisors that sum to the abundance. Prior research studied the structure of specific families of k-epns and their asymptotic density. To deepen the understanding of divisor structures in k-epns, we pose two unexplored questions. 1.   Can n have multiple redundancy sets of size k? 2.  Can n be derived from its redundancy sets?            Procedure: We propose two new properties to research these questions. 1.   Redundancy Degree: a parameter that counts the number of redundant sets of size k. 2.   Reconstructibility: A yes/no test asking whether the number n can be recovered from its size-k redundant sets by taking the least common multiple of all divisors appearing in those sets. Using these properties, we observe the density and reconstructibility of 2-epns and 3-epns amongst natural numbers through computational search. For abundant numbers of the form n = 2^ap (p is prime), we quantify redundancy degrees and reconstructibility for 2-epns and 3-epns analytically by solving simultaneous Diophantine equations for multiple redundant sets (31 proofs).          Observations: 1. As redundancy degree increases, the density of k-epns decreases, whereas the likelihood of reconstructibility increases. 2. Among 2-epns in n = 2^ap, we discover the only reconstructible Mersenne-prime based subfamily with redundancy degree of 2. 3. Among 3-npns in n = 2^ap, we show multiple subfamilies with redundancy degrees up to 3, and reconstructibility.            Conclusion: This work reveals density trends for 2-epns and 3-epns with different redundancy degrees and reconstructibility. It also uncovers new structures in n = 2^ap by applying the concepts of redundancy degree and reconstructibility.

Competition history

• CSEF 2026 Mathematical Sciences (Junior Division) · Entry J-14-02

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