相关论文: Recurrence Divisibility Tests

Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.

Here we demonstrate a sieve for analysing primes and their composites, using equivalence classes based on the modulo 6 return value as applied to the Natural numbers. Five features of this 'Hexile' sieve are reviewed. The first aspect, is…

We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of…

We present a variety of prime-generating constructions that are based on sums of primes. The constructions come in all shapes and sizes, varying in the number of dimensions and number of generated primes. Our best result is a construction…

Let $x$ and $n$ be positive integers. We prove a non-trivial lower bound for $x$, dependant only on $\omega_n$, the number of distinct prime factors of $x^n-1$. By considering the divisibility of $\varphi \mid x^n-1$ for $\varphi \mid n$,…

In this work I look at the distribution of primes by calculation of an infinite number of intersections. For this I use the set of all numbers which are not elements of a certain times table in each case. I am able to show that it exists a…

In this paper we propose several variants to perform the independence test between two random elements based on recurrence rates. We will show how to calculate the test statistic in each one of these cases. From simulations we obtain that…

Let x(n) be a recurrence relation. The main purpose of this article is to determine a recurrence for powers of x(n).

The prime numbers look like a randomly chosen sequence of natural numbers, but there is still no strict theory to determine 'Randomness'. In these years, cryptography has developed a battery of statistical tests for randomness. In this…

A classification of twin primes implies special twin primes. When applied to triplets, it yields exceptional prime number triplets. These generalize yielding exceptional prime number multiplets.

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…

The article describes prime intervals into the prime factorization of the middle binomial coefficient. Prime factors and prime powers are distributed in layers. Each layer consists of non-repeated prime numbers which are chosen (not…

The problem of N-digit sets all permutations of which give primes is discussed. Such sets may include only digits 1, 3, 7 and 9, and none of 0, 2, 5, 4, 6, 8. Direct calculations show that such full-permutation digit sets occur at N = 1, 2,…

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their…

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition,…

We initiate a systematic investigation of distribution testing in the framework of algorithmic replicability. Specifically, given independent samples from a collection of probability distributions, the goal is to characterize the sample…

We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.

The purpose of this article is to delve into the properties of invariants. The properties, explained in [2], reveal new ways to develop algorithms that allow us to test the primality of a number. In this article, some of these are shown,…

In this paper we propose necessary upper bounds for the second, third and fourth smallest prime divisors of friends of 10 based on the number of distinct prime divisors of it.