Related papers: Numbers as Data Structures: The Prime Successor Fu…
The inverse problem for representation functions takes as input a triple (X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A \subseteq X…
We have presented a multivariate polynomial function termed as factor elimination function,by which, we can generate prime numbers. This function's mapping behavior can explain the irregularities in the occurrence of prime numbers on the…
In contrast to the usual understanding of probabilistic systems as stochastic processes, recently these systems have also been regarded as transformers of probabilities. In this paper, we give a natural definition of strong bisimulation for…
We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that…
It is a challenge to manage complex systems efficiently without confronting NP-hard problems. To address the situation we suggest to use self-organization processes of prime integer relations for information processing. Self-organization…
The relation between the requirement of efficient implementability and the product state representation of numbers is examined. Numbers are defined to be any model of the axioms of number theory or arithmetic. Efficient implementability…
We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. $\left\| n \right\|$ denotes the minimum number of 1's in the expressions representing $n$. The logarithmic complexity $\left\|…
We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…
e use Prolog as a flexible meta-language to provide executable specifications of some fundamental mathematical objects and their transformations. In the process, isomorphisms are unraveled between natural numbers and combinatorial objects…
We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…
This paper introduces the Adaptive Base Representation (ABR) Theorem and proposes a novel number system that offers a structured alternative to the binary number system for digital computers. The ABR number system enables each decimal…
It is well known that physical phenomena may be of great help in computing some difficult problems efficiently. A typical example is prime factorization that may be solved in polynomial time by exploiting quantum entanglement on a quantum…
Sorting is one of the most used and well investigated algorithmic problem [1]. Traditional postulation supposes the sorting data archived, and the elementary operation as comparisons of two numbers. In a view of appearance of new processors…
The number of ordered factorizations and the number of recursive divisors are two related arithmetic functions that are recursively defined. But it is hard to construct explicit representations of these functions. Taking advantage of their…
To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…
While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…
Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously…
We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in…
This work is meant to be a step towards the formal definition of the notion of algorithm, in the sense of an equivalence class of programs working "in a similar way". But instead of defining equivalence transformations directly on programs,…
Recent work by Craig, van Ittersum, and Ono constructs explicit expressions in the partition functions of MacMahon that detect the prime numbers. Furthermore, they define generalizations, the MacMahonesque functions, and prove there are…