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 This paper revisits a method of generalization of proofs of universal sentences that was introduced by Baaz and provides several applications to the study of the factors of Fermat numbers; remarkably an improvement of his sufficient condition for a given value to be one of such factors.