Related papers: The Classical Smarandache Function and a Formula f…
In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent…
A quick way to compute generating functions related to Pell-Padovan tetranacci numbers and classical sequences of recursions of order two is provided. Eight special instances can be computed at once.
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As the application of this formula we formulate 7…
An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre…
The Smarandache function of a positive integer $n$, denoted by $S(n)$, is defined to be the smallest positive integer $j$ such that $n$ divides the factorial $j!$. In this note, we prove that for any fixed number $k > 1$, the inequality…
Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two…
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 survey the classical results on the prime number theorem
This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…
Mathematicians has been trying to prove the weak Goldbach's conjecture by adding prime numbers, as stated in the conjecture. However, we believe that the solution does not need to be analytically solved. Instead of trying to add prime…
In this paper, we study the diophantine equation ${{\sigma }_{2}}(n)-{{n}^{2}}=An+B$. We prove that except for finitely many computable solutions, all the solutions to this equation with $(A,B)=({{L}_{2m}},F_{2m}^{2}-1)$ are…
We answer a number of questions relating to the pseudo-Smarandache function Z(n). We show that the ratio of consecutive values $Z(n+1)/Z(n)$ and $Z(n-1)/Z(n)$ are unbounded; that $Z(2n)/Z(n)$ is unbounded; that $n/Z(n)$ takes every integer…
I define Goldbach counting function with N > 0 and square-free P > 0. Decomposition of this function is discovered and deduction formula is found. I propose a hypothesis on upper bound of Goldbach counting function and prove that Goldbach…
Goldbach conjecture is one of the most famous open mathematical problems. It states that every even number, bigger than two, can be presented as a sum of 2 prime numbers. % In this work we present a deep learning based model that predicts…
This paper proposes, and demonstrates the efficacy of, an elementary method for establishing a lower bound for cardinalities of selected sets of twin primes, and shows that the proof employed may be modified for selected sets of Goldbach…
The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea…
Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta,x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - \Delta^{-1}), x]$.
For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…
We obtain a simple relations for the Newman sum over multiples of a prime with a primitive or semiprimitive root 2. We consider the case of p=17 as well.
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like…