Related papers: An equivalent problem to the Twin Prime Conjecture

We consider the rooted trees which not have isomorphic representation and introduce a conception of complexity a natural number also. The connection between quantity such trees with $n$ edges and a complexity of natural number $n$ is…

Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$,…

In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of…

We proved that any even number not less than 6 can be expressed as the sum of two old primes, $2n=p_i+p_j$

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…

I present a new property of prime numbers that leads to a generalization of Cramer's conjecture. The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain…

Using the Rowland idea, we find two infinite sets of generators of primes. We also pose some conjectures concerning twin primes.

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary…

Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.

In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in…

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has…

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer $k>1$ is the sum of two figurate primes. In this paper we give an equivalent proposition to the conjecture. By considering extreme value problems with constraints…

It is shown that the unique representation of positive integers in terms of tribonacci numbers and the unique representation in terms of iterated A, B and C sequences defined from the tribonacci word are equivalent. Two auxiliary…

We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a…

We disprove the conjecture that every sufficiently large natural number $n$ is the sum of three palindromic natural numbers where one of them can be chosen to be the largest or second largest palindromic natural number smaller than or equal…

We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based…

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In a preceding paper we have proved that there exists a positive integer $K_\alpha$ such that every even integer $x > p_k^2$ can be…

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any…

The Polignac's Conjecture, first formulated by Alphonse de Polignac in 1849, asserts that, for any even number M, there exist infinitely many couples of prime numbers P, P+M. When M = 2, this reduces to the Twin Primes Conjecture. Despite…