Disproofs of two conjectures concerning nondeficient numbers

Technical Summary: Disproofs of Two Conjectures Concerning Nondeficient Numbers

Problem Statement
The paper addresses two conjectures proposed by Ross in 2024 regarding the relationship between nondeficient numbers and $\{-1, 1\}$-perfect numbers.

• A positive integer $n$ is nondeficient if $\sigma(n) \ge 2n$, where $\sigma(n)$ is the sum of divisors of $n$.
• A positive integer $n$ is $\{-1, 1\}$-perfect if its divisors $1 = d_0 < d_1 < \dots < d_k < d_{k+1} = n$ satisfy the equation $1 + \sum_{j=1}^k \lambda_j d_j = n$ for some coefficients $\lambda_j \in \{-1, 1\}$.

Ross conjectured:

1. The set of $\{-1, 1\}$-perfect numbers has the same natural density as the set of nondeficient numbers.
2. Every odd, nondeficient number that is not a perfect square is $\{-1, 1\}$-perfect.

The paper aims to disprove both conjectures by constructing specific sets of counterexamples with positive lower asymptotic density (for the first conjecture) and explicit families of counterexamples (for the second).

Methodology
The author employs analytic number theory techniques, specifically focusing on the distribution of divisors and the properties of the sum-of-divisors function $\sigma(n)$.

• Reformulation of $\{-1, 1\}$-perfectness: The paper establishes a crucial lemma (Lemma 2.1) stating that $n$ is $\{-1, 1\}$-perfect if and only if $\frac{\sigma(n) - 2n}{2}$ can be expressed as a sum of distinct proper divisors of $n$ (excluding $1$ and $n$). This transforms the problem into a subset sum problem regarding divisors.
• Density Analysis (Conjecture 1): To disprove the density conjecture, the author constructs a set $E$ of nondeficient numbers that are not $\{-1, 1\}$-perfect.
  • The strategy involves identifying integers $m$ coprime to 6 such that $\frac{\sigma(m)}{m}$ is sufficiently close to 1 (specifically, $\frac{\sigma(m)}{m} < 1 + \frac{1}{78}$).
  • Using Lemma 2.2, the author proves that the set of such integers $m$ has positive lower asymptotic density. This relies on Markov's inequality and estimates of the density of integers coprime to primorials.
  • The author then considers numbers of the form $n = 18m$. It is shown that if $m$ satisfies the density condition, $18m$ is strictly abundant (nondeficient) but fails the subset sum condition required for $\{-1, 1\}$-perfectness.
• Explicit Counterexamples (Conjecture 2): To disprove the second conjecture, the author constructs a family of odd, non-square, abundant numbers that are not $\{-1, 1\}$-perfect.
  • Let $a$ be an odd abundant square and $p$ be a prime such that $p > \sigma(a)$.
  • The number $n = ap$ is shown to be odd, non-square, and abundant.
  • By analyzing the quantity $H(n) = \frac{\sigma(n) - 2n}{2}$, the author demonstrates that $H(n)$ lies strictly between two possible sums of proper divisors of $n$, making it impossible for $n$ to be $\{-1, 1\}$-perfect.

Key Contributions and Results

1. Disproof of Conjecture 1 (Density):
   The paper proves Theorem 2.4, establishing the existence of a set $E \subseteq A \setminus P$ (where $A$ is the set of nondeficient numbers and $P$ is the set of $\{-1, 1\}$-perfect numbers) such that the lower asymptotic density $d(E) > 0$.

   • Result: Since $P \subseteq A$ and $E$ is a disjoint subset of $A$ with positive density, it follows that $d(P) < d(A)$. This definitively disproves Ross's conjecture that the densities are equal.

2. Disproof of Conjecture 2 (Odd Non-Squares):
   The paper proves Theorem 3.1, providing an explicit construction of counterexamples.

   • Result: For any odd abundant square $a$ and a prime $p > \sigma(a)$, the number $n = ap$ is an odd, non-square, abundant number that is not $\{-1, 1\}$-perfect. This disproves the conjecture that all such numbers are $\{-1, 1\}$-perfect.

Significance
The paper's primary significance lies in correcting the understanding of the distribution and structural properties of $\{-1, 1\}$-perfect numbers relative to nondeficient numbers.

• It clarifies that the class of $\{-1, 1\}$-perfect numbers is a "thin" subset of nondeficient numbers in terms of natural density, contrary to the intuition suggested by Ross's first conjecture.
• It demonstrates that the property of being $\{-1, 1\}$-perfect is not guaranteed even for odd, non-square abundant numbers, refuting the second conjecture.
• The work contributes to the broader study of perfect number variants by providing rigorous density arguments and explicit arithmetic constructions that limit the scope of these generalized perfect number definitions.

The author notes that the development of these proofs was supported by interactions with GPT-5.5 Pro during the exploratory stages, though all mathematical content was independently verified and revised by the author. The results rely on standard number-theoretic tools, including properties of the Euler totient function, primorials, and the distribution of integers with specific divisor sum properties.