Abstract We show that the class of Kalmár elementary functions can be inductively generated from the addition, the integer remainder and the base-two exponentiation, hence improving previous results by Marchenkov and Mazzanti. We also prove that the substitution basis defined by these three operations is minimal. Furthermore, we discuss alternative substitution bases under arity constraints.

Abstract We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the $ p $-adic valuation, the number-of-divisors function, the sum-of-divisors function, Euler’s totient function, the modular inverse, the integer part of the root, the integer part of the logarithm, the multiplicative order and the discrete logarithm. Although these are very complicated, they only involve elementary operations, and, to our knowledge, no other closed form of this kind is known for the aforementioned functions. AMS Subject Classification: 11A25 (primary), 03D20, 03D55

Abstract We show that the set of natural numbers has an exponential diophantine definition in the rationals. It follows that the corresponding decision problem is undecidable.

Abstract A structure of finite signature is constructed so that: for all existential formulas and for all tuples of elements of the same length as the tuple one can decide in a quadratic time depending only on the length of the formula, if holds in the structure. In other words, the structure satisfies the relativized model-theoretic version of P=N P in the sense of [4]. This is a model-theoretical approach to results of Hemmerling and Gaßner.

Abstract We give a model-theoretic proof of the fact that for all infinite Abelian groups P ≠ NP in the sense of binary nondeterminism. This result has been announced 1994 by Christine Gaßner.

Abstract We use a new version of the Definability Theorem of Beth in order to unify classical theorems of Yuri Matiyasevich and Jan Denef in one structural statement. We give similar forms for other important definability results from Arithmetic and Number Theory.