Geodesic
Solvability of the theory of integers with addition, order, and multiplication by an arbitrary number
Matematičeskie zametki, Tome 13 (1973) no. 5, pp. 667-675 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is known that the arithmetic of natural and integer numbers is unsolvable. Even the universal theory of integers with addition and multiplication is unsolvable. It is proved herein that an elementary theory of integers with addition, order, and multiplication by one arbitrary number is solvable and multiplication by the power of one number is unsolvable. For a certain $n$, the universal theory of integers with addition and $n$ multiplications by an arbitrary number is also unsolvable. 
@article{MZM_1973_13_5_a4,
     author = {Yu. G. Penzin},
     title = {Solvability of the theory of integers with addition, order, and multiplication by an arbitrary number},
     journal = {Matemati\v{c}eskie zametki},
     pages = {667--675},
     year = {1973},
     volume = {13},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_5_a4/}
}
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Yu. G. Penzin. Solvability of the theory of integers with addition, order, and multiplication by an arbitrary number. Matematičeskie zametki, Tome 13 (1973) no. 5, pp. 667-675. http://geodesic.mathdoc.fr/item/MZM_1973_13_5_a4/