Related papers: On Cross Hyperoperatorial Migration of Properties,…
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization…
Literature considers under the name \emph{unimaginable numbers} any positive integer going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathematical…
Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with…
The symbolic representation of a number should be considered as a data structure, and the choice of data structure depends on the arithmetic operations that are to be performed. Numbers are almost universally represented using position…
Each natural number can be associated with some tree graph. Namely, a natural number $n$ can be factorized as $$ n = p_1^{\alpha_1}\ldots p_k^{\alpha_k},$$ where $p_i$ are distinct prime numbers. Since $\alpha_i$ are naturals, they can be…
We study some essential arithmetic properties of a new tree-based number representation, {\em hereditarily binary numbers}, defined by applying recursively run-length encoding of bijective base-2 digits. Our representation expresses giant…
Prime factorization is an outstanding problem in arithmetic, with important consequences in a variety of fields, most notably cryptography. Here we employ the intriguing analogy between prime factorization and optical interferometry in…
Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number…
In this paper we give an attempt to extend some arithmetic properties such as multiplicativity, convolution products to the setting of operators theory. We provide a significant examples which are of interest in number theory. We also give…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…
We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct…
The discrete Fourier transform of the greatest common divisor is a multiplicative function, if taken with respect to the same order of the primitive root of unity, which is a well known fact. As such, the transform can be expressed in the…
The integer $d$ is called an exponential divisor of $n=\prod_{i=1}^r p_i^{a_i}>1$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i \mid a_i$ for every $1\le i \le r$. The integers $n=\prod_{i=1}^r p_i^{a_i}, m=\prod_{i=1}^r p_i^{b_i}>1$ having…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^(n+1) (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a…
The tree based representation described in this paper, hereditarily binary numbers, applies recursively a run-length compression mechanism that enables computations limited by the structural complexity of their operands rather than by their…
We study a bijective map from integer partitions to the prime factorizations of integers that we call the "supernorm" of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors…
A notion of super operator system is defined which generalizes the usual notion of operator systems to include certain unital involutive operator spaces which cannot be represented completely isometric as a concrete operator system on some…
Obtained a new property of superposition of the generating functions ln(1/(1-F(x))), where F(x) - generating function with integer coefficients, which allows the construction a primality tests. The theorem which is based on compositions of…
Can we do arithmetic in a completely different way, with a radically different data structure? Could this approach provide practical benefits, like operations on giant numbers while having an average performance similar to traditional…