English

Disproofs of two conjectures concerning nondeficient numbers

Number Theory 2026-07-13 v1

Abstract

A positive integer nn is said to be nondeficient if σ(n)2n\sigma(n) \geq 2n. Letting the positive divisors of a positive integer nn be written as 1=d0<d1<<dk<dk+1=n1 = d_0 < d_1 < \cdots < d_k < d_{k+1} = n, and letting S\mathcal{S} denote a set of integers, if there exist values λjS\lambda_j \in \mathcal{S} such that 1+j=1kλjdj=n1 + \sum_{j=1}^{k} \lambda_j d_j = n, then nn is said to be an S\mathcal{S}-perfect number. Ross, in 2024, introduced the study of S\mathcal{S}-perfect numbers, and concluded with two conjectures that each concern both {1,1}\{ -1, 1 \}-perfect numbers and nondeficient numbers. We disprove both of these conjectures.

Keywords

Cite

@article{arxiv.2607.11043,
  title  = {Disproofs of two conjectures concerning nondeficient numbers},
  author = {John M. Campbell},
  journal= {arXiv preprint arXiv:2607.11043},
  year   = {2026}
}

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