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The predicate of divisibility on two consecutive numbers DW(x,y) ÷ x | y ∧ 1+x | y was introduced by L. van den Dries and A. Wilkie when they studied some properties of subsets of natural numbers, existentially definable with unit, addition and divisibility. Undecidability of the existential theory of natural numbers with multiplication and DW and definability of addition and multiplication in terms of DWand divisibility is proved in the paper. Then the definability of multiplication in the terms of addition and DW is proved. Some definability questions for order and DW are also considered in the paper.

Some Decidability and Definability Problems for the Predicate of the Divisibility on Two Consecutive Numbers