IJRR

International Journal of Research and Review

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Review Paper

Year: 2021 | Month: May | Volume: 8 | Issue: 5 | Pages: 379-388

DOI: https://doi.org/10.52403/ijrr.20210547

The Equation of the Set of Natural Numbers Just to Sum

Tulus Nadapdap1, Tulus2, Opim Salim3

1Posgraduate Students Faculty of Mathematics and Natural Sciences Universitas Sumatera Utara, Indonesia
2,3Postgraduate Lecturer Faculty of Mathematics and Natural Sciences Universitas Sumatera Utara, Indonesia

Corresponding Author: Tulus Nadapdap

ABSTRACT

Systems of equations of the form X = Y + Z and X = C, in which the unknowns are sets of integers,”+” denotes pairwise sum of sets S + T = m + n m S, n T , and C is an ultimately periodic constant. When restricted to sets of natural numbers, such equations can be equally seen as language equations over a one-letter alphabet with concatenation and regular constants, and it is shown that such systems are computationally universal, in the sense that for every recursive set S N there exists a system with a unique solution containing T with S = n 16n + 13 T. For systems over sets of all integers, both positive and negative, there is a similar construction of a system with a unique solution S = {n|16n ∈ T} representing any hyper-arithmetical set S ⊆ N.

Keywords: Language equations, Natural numbers, Equations of natural number.

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