Related papers: La cometa di Goldbach e ... le altre
The Collatz function is defined as C(n) = n / 2 if n is even and C(n) = 3n + 1 if n is odd. The Collatz conjecture states that every sequence generated by the Collatz function ends with the cycle (4, 2, 1) after a finite number of…
In this paper we propose an alternative formulation of the binary and ternary Goldbach conjectures as the systems of equations involving the Euler $\phi$-function.
According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers : $n = p + q$. We construct a network where each node is a prime number and corresponding to every even number $n$, we put a link between the…
In this paper, we investigate exceptional sets in the Waring-Goldbach problem for unlike powers. For example, estimates are obtained for sufficiently large integers below a parameter subject to the necessary local conditions that do not…
Let Omega(n) be the volume of the unit ball in R^n. We formulate as an infinite product the gamma function ratio gamma(x+1/2)/gamma(x),x>0, which allows us to reproduce and /or produce a variety of formulas and inequalities, some of them…
It is well known that every positive integer N can be written as the sum of non-consecutive powers of the golden ratio. We prove that the non-positive powers, together with the parity of the first positive power, can determine the positive…
In this paper, we develop the method of circle of partitions and associated statistics. As an application we prove conditionally the binary Goldbach conjecture. We develop a series of steps to prove the binary Goldbach conjecture in full.…
For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices $1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all Hamiltonian…
In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper, we obtain explicit numerical estimates for the…
Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…
Let $\Lambda\left(n\right)$ be the von Mangoldt function and $r_{G}\left(n\right) := \sum_{m_1 + m_2=n} \Lambda \left(m_1 \right) \Lambda\left(m_2 \right)$ be the counting function for the numbers that can be written as sum of two primes…
In this paper, we show that every pair of large even integers satisfying certain necessary conditions can be expressed as a pair of one prime, one prime square, two prime cubes and 56 powers of 2.
The famous Goldbach conjecture remains open for nearly three centuries. Recently Goldbach graphs are introduced to relate the problem with the literature of Graph Theory. It is shown that the connectedness of the graphs is equivalent to the…
Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum of euler's totient function from k equal 1 to n whene p divides n and p…
The Collatz map is defined for a positive even integer as half that integer, and for a positive odd integer as that integer threefold, plus one. The Collatz conjecture states that when the map is iterated the number one is eventually…
The purpose of this paper is to show three general formulas of three global characteristic coefficients of Collatz function. The Collatz function is defined by the following operation on an arbitrary positive integer if N is odd multiply it…
This article is a survey based on our earlier paper ("The 'Vertical' Generalization of the Binary Goldbach's Conjecture as Applied on 'Iterative' Primes with (Recursive) Prime Indexes (i-primeths)" [11]), a paper in which we have proposed a…
A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the positive divisors of $n$. In this paper, we establish the structure theorem of odd multiperfect…
The Collatz Conjecture can be stated as: using the reduced Collatz function $C(n) = (3n+1)/2^x$ where $2^x$ is the largest power of 2 that divides $3n+1$, any odd integer $n$ will eventually reach 1 in $j$ iterations such that $C^j(n) = 1$.…
For a fixed rational number g different from -1,0,1 and integers a and d the set N_g(a,d) of primes p for which the order of g(mod p) is congruent to a(mod d) is considered. It is shown, assuming the Generalized Riemann Hypothesis (GRH),…