相关论文: The Carmichael numbers up to $10^{17}$

In this article we show that the Czech mathematician Vaclav Simerka discovered the factorization of (10^17-1)/9 using a method based on the class group of binary quadratic forms more than 120 years before Shanks and Schnorr developed…

We consider the integers having the property of reversing when multiplied by a specific integer k. First, we proved that k should be either 1, 4 or 9. Second, we classify these integers as (10, 1)- reverse multiples, (10, 4)- reverse…

The Jacobsthal function has aroused interest in various contexts in the past decades. We review several algorithmic ideas for the computation of Jacobsthal's function for primorial numbers and discuss their practicability regarding…

The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…

The paper describes a prime factorization of the Catalan numbers. Odd prime factors are distributed in layers in accordance with Legendre's formula. The content of each layer is a network of the intervals, Chebyshev's Segments. The primes…

There are 123,650 partial groups of order at most 9 and 178,937,003 partial groups of order 10. We explain a computer enumeration of these results and provide a complete list of indecomposable partial groups of order at most 5. We also…

In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…

We generalize the definition of spoof perfect numbers to multiperfect numbers and study their characteristics. As a result, we find several new odd spoof multiperfect numbers, akin to Descartes' number. An example is $8999757$, which would…

Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to…

Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). The recovered period for the sequence of the first 10000 prime numbers is equal to…

We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits…

For $a \neq 1$ and $p$ prime, we define numbers of the form $pa^2$ to be Square-Prime (SP) Numbers. For example, 75 = 3 $\cdot$ 25; 108 = 3 $\cdot$ 36; 45 = 5 $\cdot$ 9. These numbers are listed in the OEIS as A228056. We study the…

Suppose that $\alpha,\beta\in\mathbb{R}$. Let $\alpha\geqslant1$ and $c$ be a real number in the range $1<c< 12/11$. In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski--Shapiro sequence, which…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

This work is meant to demonstrate new class of prime numbers -- cyclic prime numbers, that can be derived from any prime number at certain numeric systems. Cyclic prime numbers are also related to the cyclic numbers and full reptend prime…

We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known…

We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer $n$, which is not a prime power, is an elliptic Carmichael number for a random curve…

I give some claims on primorial prime numbers for interested readers in number theory.

The prime number 357686312646216567629137 is notable because of the unusual property that it remains prime successively on removing the left digit until there are no remaining digits. We explore here the distributions of the number of left…

A modified Lagrange Polynomial is introduced for polynomial extrapolation, which can be used to estimate the equally spaced values of a polynomial function. As an example of its application, this article presents a prime-generating…