Related papers: Finding Large Primes
We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…
Given a (possibly infinite) subset $A$ of the natural numbers, we ask how many times a fair six-sided die must be rolled until the rolled numbers add up to an element of $A$. Using a one-dimensional dynamic programming recursion together…
Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most $\frac 14$ of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius…
We investigate the probability that a random odd composite number passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to $2^{200}$, our results improve on prior…
The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the…
For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log…
In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a…
We propose a divisibility test for all integers which have 1, 3, 7 or 9 in their unit's place. In particular, then, the test applies for all prime divisors except 2 and 5.
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…
The strong Lucas test is a widely used probabilistic primality test in cryptographic libraries. When combined with the Miller-Rabin primality test, it forms the Baillie-PSW primality test, known for its absence of false positives,…
The main result of this thesis is to show that there are only finitely many integers $n$ such that both $n$ and $d(n)$ are highly composite numbers at the same time, where $d(n)$ is the divisor function. Bertrand's postulate [4] is used…
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 primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either…
Simple divisibility rules are given for the 1st 1000 prime numbers.
We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of…
In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as…
We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we…
In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…
We show that if you represent all primes with less than n-digits as points in n-dimensional space, then they can be stored and retrieved conveniently using n-dimensional geometry. Also once you have calculated all the prime numbers less…
We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…