Geodesic
Decidability of the universal theory of natural numbers with addition and divisibility
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VII, Tome 60 (1976), pp. 15-28

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The class of all quantifier-free formulas constructed from atomic formulas of the form $(x+y=z)$, $(x=1)$, and $(x/y)$ is considered, where the predicate symbol “|” is interpreted as the divisibility relation on nonnegative integers. The decidability isproved of the set of all formulas of this form which are true for at least one choice of values for the variables. This result is equivalent to the decidability of the universal theory of natural numbers with addition and divisibility. 
@article{ZNSL_1976_60_a1,
     author = {A. P. Beltiukov},
     title = {Decidability of the universal theory of natural numbers with addition and divisibility},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {15--28},
     publisher = {mathdoc},
     volume = {60},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_60_a1/}
}
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A. P. Beltiukov. Decidability of the universal theory of natural numbers with addition and divisibility. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VII, Tome 60 (1976), pp. 15-28. http://geodesic.mathdoc.fr/item/ZNSL_1976_60_a1/