Related papers: Prime number generation and factor elimination

We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.

The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

We discuss a method for computing the generating function for the multiplicity distribution in field theories with strong time dependent external sources. At leading order, the computation of the generating function reduces to finding a…

For a class of Lucas sequences ${x_n}$, we show that if $n$ is a positive integer then $x_n$ has a primitive prime factor which divides $x_n$ to an odd power, except perhaps when $n = 1, 2, 3 or 6$. This has several desirable consequences.

A general explicit form for generating functions for approximating fractional derivatives is derived. To achieve this, an equivalent characterisation for consistency and order of approximations established on a general generating function…

Instances generation is crucial for linear programming algorithms, which is necessary either to find the optimal pivot rules by training learning method or to evaluate and verify corresponding algorithms. This study proposes a general…

A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of…

In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some…

We present, (partially) analyze, and apply an efficient algorithm for the simulation of multivariate Pareto records. A key role is played by minima of the record-setting region (we call these generators) each time a new record is generated,…

Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an…

Let $f(x)\in \mathbb{F}_q[x]$ be an irreducible polynomial of degree $m$ and exponent $e$, and $n$ be a positive integer such that $\nu_p(q-1)\ge \nu_{p}(e)+\nu_p(n)$ for all $p$ prime divisor of $n$. We show a fast algorithm to determine…

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…

We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm…

In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.

Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in…

We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, and $\zeta$. Particularly,…

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers.…

The prime number problem falls within the realm of number theory, specifically elementary number theory. Current research approaches have unnecessarily complicated this matter. In contrast to more advanced mathematical tools, the methods of…

The theorem below gives another way of computing the distribution prime counting function without using recursion and the values of Prime numbers