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On the first-order theory of the remainder

Logic 2025-06-05 v1

Abstract

It is proved that the first-order theory of the structure (N,mod) is undecidable. Here mod denotes the operation of computing the remainder for any division between positive integers; i.e. x mod y is the remainder obtained by the division x : y.

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Cite

@article{arxiv.2506.03204,
  title  = {On the first-order theory of the remainder},
  author = {Mihai Prunescu},
  journal= {arXiv preprint arXiv:2506.03204},
  year   = {2025}
}

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2 pages