Kajetan Młynarski | Searcharxiv

arXiv2018

Notes on properties of binary strings which encode prime occurrences

A binary string representation of prime occurrences is a sequence of bits, where $1$ entries encode positions of prime numbers. This is a convenient representation for analysis of prime distribution, since it allows for application of a broad range of existing string-analysis algorithms to problems in number theory. Binary strings of prime occurrences can be also generated with simple algorithms. Here we discuss three such algorithms and we demonstrate their applicability using the example of proving Goldbach's hypothesis for some limited sets of even numbers. This work formulates three open questions (conjectures) regarding the distribution of primes.

math.NT↗

arXiv2011

Reflexivity and the diagonal argument in proofs of limitative theorems

This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. Gödel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-Gödel's theorem). The fact, that a formal system contains a sentence, which introduces reflexitivity, does not imply, that the same system does not contain a sentence or a proof procedure which solves this problem. Second basic method of proof - diagonal argument (i.e. showing non-eqiunumerosity of a program set with the set of real numbers) does not exclude existance of a single program, capable of computing all real numbers. In this work, we suggest an algorithm generating real numbers (arbitrary, infinite in the limit, binary strings), and we speculate it's meaning for theoretical computer science.

cs.LO↗