conjecture

In this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the same pattern: 1) Upward trajectory. Odd and even numbers alternate until the cycle reaches an upper bound 2) Downward trajectory. Two or more consecutive even numbers follow until it reaches another odd number. At this point, it's the beginning of the following cycle. Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle. Finally, we prove that both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a...

The Collatz’s conjecture is an unsolved problem in mathematics. It is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse conjecture or problem [1, 2, 3]. Take any positive integer n. If n is even then divide it by 2, else do ”triple plus one” and get 3n+ 1. The conjecture is that for all numbers, this process converges to one. In the modular arithmetic notation, define a function f as follows:

Gauss's last diary entry concerning the number of solutions of x^2 + y^2 + x^2 y^2 == 1 mod p (p. 293) Fermat's little theorem (p. 70) a (n(p) – 1) = 1 mod p (p. 8) p = x^2 + y^2 p = x^2 + 2 y^2 p = x^2 + 3 y^2 Let a^(n(p) – 1) == p + x^2 y^2 Richard Crandal and Carl Pomerance. Prime Numbers: A Computational Perspective, c. 2005 (p. 321) Definition of elliptical curve y^2 = x^3 + ax + b, let a=b=c y^2 = x^3 Then Let p^(n(p) – 1) == p + x^2 y^2 p(n(p) – 1) == p + x^2 x^3 {2,3,.. p} 2^(1 – 1) == 2 + x^5 0 == 1 + x^5, x =-1 3^(2 – 1) == 3 + x^5

Clayton and Stevens argue that political liberals should engage with the religiously unreasonable by offering religious responses and showing that their religious views are mistaken, instead of refusing to engage with them. Yet they recognize that political liberals will face a dilemma due to such religious responses: either their responses will alienate certain reasonable citizens, or their engagements will appear disingenuous. Thus, there should be a division of justificatory labour. The duty of engagement should be delegated to religious citizens. In this comment, I will argue that the division of justificatory labour is indefensible. This dilemma can be avoided if politicians and political philosophers correctly use conjecture, a form of discourse that involves non-public reason. As a conditional response, conjecture avoids alienating any reasonable citizens. Also, if conjecture is given in a sincere and open-minded manner, then the problem of disingenuousness can be overcome. My comment concludes that while the engagement of politicians and political philosophers does not necessarily jeopardize overlapping consensus, they should be permitted, or perhaps even required, to engage with the religiously unreasonable due to the natural duty of justice.

The strong cosmic censorship conjecture, whose validation asserts the deterministic nature of general relativity, has been studied for charged BTZ black holes in three dimensional general relativity, as well as for Nth order pure Lovelock gravity in d=2N+1 spacetime dimensions. Through both analytical and numerical routes, we have computed the ratio of the imaginary part of the quasi-normal mode frequencies with the surface gravity at the Cauchy horizon. The lowest of which corresponds to the key parameter associated with violation of strong cosmic censorship conjecture. Our results demonstrate that this parameter is always less than the critical value (1/2), thereby respecting the strong cosmic censorship conjecture. This is in complete contrast to the four or, higher dimensional black holes, as well as for rotating BTZ black hole, where the violation of strong cosmic censorship conjecture exists. Implications and possible connection with the stability of the photon orbits have been discussed.

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Sch\"onheim conjectured that if $\{H_i\alpha_i\}_{i=1}^{i=s}$, $\alpha_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. We consider the Herzog-Sch\"onheim conjecture for free groups of finite rank and propose a new approach, based on an extension of the Davenport-Rado result for $G=\mathbb{Z}$.

We show how the problems related to the Collatz map $T$ can be transferred to the language of functional analysis. We associate with $T$ certain linear operator $\mathcal{T}$ and show that cycles and (hypothetical) diverging trajectory (generated by $T$) correspond to certain classes of fixed points of operator $\mathcal{T}$. Furthermore, we demonstrate connection between dynamical properties of operator $\mathcal{T}$ and map $T$. We prove that absence of nontrivial cycles of $T$ leads to hypercyclicity of operator $\mathcal{T}$. In the end we construct an invariant measure for $\mathcal{T}$ in a slightly different setup.

In this paper, we consider the abc conjecture. Assuming that c<rad2(abc) is true, we give the proof of the abc conjecture for \epsilon is positive,  then for the case \epsilon \in ]0,1[, we consider that the abc conjecture is false, from the proof, we arrive in a contradiction.

In this paper we have given an algorithmic proof of an long standing Barnette&#39;s conjecture (1969) that every 3-connected bipartite cubic planar graph is hamiltonian. Our method is quite different than the known approaches and it rely on the operation of opening disjoint chambers, bu using spiral-chain like movement of the outer-cycle elastic-sticky edges of the cubic planar graph. In fact we have shown that in hamiltonicity of Barnette graph a single-chamber or double-chamber with a bridge face is enough to transform the problem into finding specific hamiltonian path in the cubic bipartite graph reduced. In the last part of the paper we have demonstrated that, if the given cubic planar graph is non-hamiltonian then the algorithm which constructs spiral-chain (or double-spiral chain) like chamber shows that except one vertex there exists (n-1)-vertex cycle.

The paper begins with a reference to Riemann's hypothesis on the sequence of prime numbers, still unproven today, and goes on to illustrate the twin prime conjecture and the more general Polignac's conjecture; we then recount the recent result by Yitang Zhang about it, and the improvements obtained thanks to the online mathematical collaboration called Polymath 8.

In this paper, we investigate the possible scenarios in which a number does not satisfy the Collatz Conjecture. Specifically, we examine numbers which may have a looping Collatz reduction sequence as well as numbers which may lead to a diverging Collatz reduction sequence. In order to investigate these, we look at the parity of the numbers in a general Collatz reduction sequence. Further, we examine cases in which these parity cycles repeat themselves infinitely in the reduction sequence. Through the research conducted in the paper, we formulate a necessary condition for looping in the Collatz Conjecture. We also prove that if a number has a diverging reduction sequence, then it must generate an infinite non-repeating parity cycle.

The Birch and Swinnerton-Dyer Conjecture is a well known mathematics problem in the area of Elliptic Curve. One of the crowning moments is the paper by Andrew Wiles which is difficult to understand let alone to appreciate the conjecture. This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field of rational numbers. Then we present major results like the theorems of Mordell and Mazur leading us to the current state of the conjecture.

The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatz conjecture.

A conjecture of Moore claims that if Γ is a group and H a finite index subgroup of Γ such that Γ − H has no elements of prime order (e.g. Γ is torsion free), then a Γ-module which is projective over H is projective over Γ. The conjecture is known for finite groups. In that case, it is a direct consequence of Chouinard's theorem which is based on a fundamental result of Serre on the vanishing of products of Bockstein operators. It was observed by Benson, using a construction of Baumslag, Dyer and Heller, that the analog of Serre's Theorem for infinite groups is not true in general. We prove that the conjecture is true for groups which satisfy the analog of Serre's theorem. Using a result of Benson and Goodearl, we prove that the conjecture holds for all groups inside Kropholler's hierarchy LHF , extending a result of Aljadeff, Cornick, Ginosar, and Kropholler. We show two closure properties for the class of pairs of groups (Γ, H) which satisfy the conjecture, the one is closure under morphisms, and the other is a closure operation which comes from Kropholler's construction. We use this in order to exhibit cases in which the analog of Serre's theorem does not hold, and yet the conjecture is true. We will show that in fact there are pairs of groups (Γ, H) in which H is a perfect normal subgroup of prime index in Γ, and the conjecture is true for (Γ, H). Moreover, we will show that it is enough to prove the conjecture for groups of this kind only.

The Ratios Conjecture of Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] predicts the answers to numerous questions in number theory, ranging from n-level densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture gives a recipe to generate these answers, which are believed to be correct up to square-root cancelation. These predictions have been verified, for suitably restricted test functions, for the 1-level density of orthogonal [HuyMil, Mil5, MilMo] and symplectic [Mil3, St] families of Lfunctions. In this paper we verify the conjecture's predictions for the unitary family of all Dirichlet L-functions with prime conductor; we show square-root agreement between prediction and number theory if the support of the Fourier transform of the test function is in (−1, 1), and for support up to (−2, 2) we show agreement up to a power savings in the family's cardinality.

The distinction between the Domain of Natural Numbers and the Domain of Line gets highlighted. This division provides the new perception to the Fermat’s Conjecture, where to place it and how to prove it. The reasons why the Fermat’s Conjecture remained unproven for 382 years are examined. The Fermat’s Conjecture receives the proof in the Domain of Natural Numbers only. The equation a n + b n = c n with positive integers a, b, c, n is not the Fermat’s Conjecture in the Domain of Line.

Let (x – a) (x – b) = (x^2 – (a+b) x + ab) George E. Andrews. Number Theory, paper, c. 1971. (p. 62) Theorem 5-3: The congruence (m-1)! ==-1 (mod m) holds if and only if m is a prime. Then, m! ==-m (mod m) Let (0 – (a+b) 0 + ab) = m! ==-m (mod m) That is, any m! , where m is prime, may be factored into a times b. For example, m = { 2,3,5,7, … p } 2! = 2*1 ==-2 (mod 2) 3! = 3*2*1 = 3*2 ==-3 (mod 3) 5! = 5*4*3*2*1 = 60*2 ==-5 (mod 5) 7! = 7*6*5*4*3*2*` = 42*20*6 =-7 (mod 7) Corollary 5.2: (Fermat's Little Theorem, Theorem 3-4) (p. 36) n^p == n (mod p) n^(p-1) == 1 (mod p) (m-1)! ==-1 (mod m) (m-1)! == n^(p-1) (mod p) (m-1)!-n^(p-1) == 0 (mod p)