Related papers: The Farey Sieve
A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
We explore a new sieve that generates both primes and prime factorizations, without resorting to division. We demonstrate that the integer sequences generated by the sieve are the p-adic valuations of n, and that each is a fractal sequence.…
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
We consider a certain linear recursive relation with integer parameters and study some of its algebraic and geometric properties, with the purpose of estimating the number of chains of valences in the Farey series.
This article develops a new sieve method which by adding an additional axiom to the classical formulation breaks the well-known parity problem and allows one to detect primes in thin, interesting integer sequences. In the accompanying paper…
This paper presents a distinctive prime detection approach. This method use GM-(n+1) sequences to effectively eliminate complex numbers. The sequences, which consist of odd a number of (n+1), exclude all components except for the initial…
In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.
The classical Farey sequence of height $Q$ is the set of rational numbers in reduced form with denominator less than $Q$. In this paper we introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the…
While it is trivial to multiply two C-finite sequences (just like integers), it is not quite so trivial to "factorize" them, or to decide whether they are "prime". The former is plain linear algebra, while the latter is heavy-duty…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein together with theorems corollaries, formulae, examples, mathematical criteria, etc. (about integer sequences, numbers, quotients, residues,…
By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).
In this paper we consider a slightly different sieve method from Eratosthenes' to get primes. We find the periodicity and mirror symmetry of the pattern.
We generalize classical results on the gap distribution (and other fine-scale statistics) for the one-dimensional Farey sequence to arbitrary dimension. This is achieved by exploiting the equidistribution of horospheres in the space of…
We prove some Hardy-type inequalities via an approach that involves constructing auxiliary sequences.
Ferrers diagrams are used to visually represent integer partitions. We describe a way to use Ferrers diagrams to uniquely represent integers in terms of their prime factors. This leads to a lower bound on the number of primes less than a…
We propose a multi-scale analysis method for studying arithmetic properties of integer sets, such as primality. Our approach organizes information through a hierarchy of nested sequences, where each level enables a hierarchical expression…
In this paper, we introduced the theory of the sieve function transformation. Using the principle of sieve function transformation, we improved sieve method, and obtained the difference range of similar sieve function values. For this, we…
Suppose P is a set of primes, such that for every p in P, every prime factor of p-1 is also in P. If P does not contain all primes, we apply a new sieve method to show that the counting function of P is O(x^{1-c}) for some c>0, where c…
The Arithmetic Fourier Transform is a numerical formulation for computing Fourier series and Taylor series coefficients. It competes with the Fast Fourier Transform in terms of speed and efficiency, requiring only addition operations and…