Collatz Conjecture - The Proof (original) (raw)
Collatz conjecture states that : start with any positive integer n. If the integer is odd, multiply it by 3 and add 1. If the number is even, keep on dividing it by 2 until an odd integer is obtained. After repeating this sequence again and again, one will be obtained as the final result. It is also known as the 3n+1 problem or the Ulam conjecture. This paper presents the proof of the collatz conjecture using the basic concepts of number theory.
The Proof of the Collatz Conjecture
viXra, 2017
number. It is proved that by repeating operations for the function () with any natural , completes with the unit (one), it follows that the Collatz Conjecture is true and it has been proved.
The only five expressions of numbers which respect the Collatz conjecture
This is a mathematical letter that uses easy tools in order to prove some interesting results about the Collatz conjecture. This short article carries on the efforts of the previous article about the Collatz conjecture which proves easily an interesting useful sequence with some other results by following a logical method. This work demonstrates also an interesting result which is the five only expressions of the numbers which respect the Collatz conjecture. This result may simplify many other demonstrations regarding the Collatz conjecture. Hence, I invite the readers to discover this content in order to make a new step towards the proof of this beautiful useful conjecture.
Simple arithmetic statements equivalent to the proof of the Collatz conjecture
Let's stay optimistic and believe that the logic and the modern mathematical tools can be enough in order to develop solutions for any mathematical problem, especially the Collatz conjecture. We could use previous published short works in making a new approach in the study of the Collatz conjecture. This article proposes new results with easy tools and gives simple arithmetic statements which are equivalent to the proof of the famous Collatz conjecture. I hope that this article helps the lovers of sequences and number theory in understanding new aspects of this conjecture and I hope that even the scientists who only apply mathematics in their fields of study can use the results of this work.
Solution to the Collatz Conjecture
Solution to the Collatz Conjecture
The Collatz or 3x + 1 conjecture is perhaps the simplest stated yet unsolved problem in mathematics in the last 70 years. It was circulated orally by Lothar Collatz at the International Congress of Mathematicians in Cambridge, Mass, in 1950 (Lagarias, 2010). The problem is known as the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem. In this concise paper I provide a proof of this conjecture, by finding an upper bound to the Collatz sequence and, as a consequence, a contradiction.
Proof of the Collatz conjecture
In this research a short proof of the Collatz conjecture is presented. The proof will begin with a proposed initial positive odd integers form and the number of steps and process by which it eventually reaches the number one will be investigated. Thus it will be shown that every n has a well-defined stopping time.
The Collatz Conjecture A PROPOSAL FOR ITS SOLUTION
The Collatz Conjecture A PROPOSAL FOR ITS SOLUTION, 2024
The purpose of this paper is to give a proposal for solving the Collatz conjecture. If the conjecture is taken to refer to finite integer positive numbers however large and finite sets however large, without reference to infinity, then the conjecture is proved to be valid.
Complete Proof of the Collatz Conjecture
2021
The Collatzβs 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 βtriple plus oneβ 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:
Collatz's conjecture from an elementary point of view -Ardito prof. Nicola
Collatz Conjecture Proof, 2025
One of the problems of elementary mathematics, still unsolved, is the Collatz conjecture (after the mathematician Lothar Collatz). It concerns the following function between natural numbers: f: N βN N/2 if n is even f(N)= 3*N+1 if n is odd Iterating the function several times after a certain number of iterations gives the number 1 and continuing the iterations gives the cycle 1-4-2-1. Then the initial number N, in subsequent iterations, is multiplied by 3 (adding 1) and divided by 2 so that it is a succession of odd steps (when multiplying by 3 and adding 1) and even passes (when dividing by 2) which, after a certain total number of even passes bi and a certain total number of odd steps ai, converges to 1 based on the relation π π π * π΅ * πΉππ π π π π = π where is the final residue which is between πΉππ π 1.1 and 1.25 for reference [1] and therefore we will have π π π * π΅ π π π < π. In this article, a new conjecture equivalent to Collatz's canonical conjecture is presented. In fact, in the following the Collatz conjecture customized by me will be applied in a faster way considering only natural numbers N modulo 4 and that the sequence of these numbers (after an initial number that can also be even mod. 4) is reduced to the sequence of only odd numbers β‘ 1 mod. 4 and β‘ 3 mod. 4. This way of proceeding leads to the same results as the canonical Collatz succession. We will indicate below with ak the total of odd passes and with bk the total of even passes obtained in a certain number k of successive iterations and we will indicate with N[10] the value of the number N after 10 total passes both odd and even. In addition, the term "double passage" will indicate the passage of N to π * π΅+π π that is, the odd passage always followed by an even passage. The odd numbers N β‘ 1 mod. 4 and N β‘ 3 mod. 4 will be indicated later also and respectively with N=4*h0 +1 and N=4*h0 +3 and their subsequent ones in the Collatz sequence will be indicated with N1=4*h1+1 and N1=4*h1+3 ....... Nn=4*hn+1 and Nn=4*hn+3 where h0, h1, h2,...., hn,... we will call the coefficients of 4 of the respective odd numbers N, N1, N2,.....,Nn... β‘ 1 mod. 4 and β‘ 3 mod. 4. Also, I will later call all even numbers N β‘ 0 mod. 4 numbers of type 1, all even numbers N β‘ 2 mod. 4 numbers of type 2, all odd numbers N β‘ 1 mod. 4 numbers of type 3 and all odd numbers N β‘ 3 mod. 4 numbers of the type 4.
The objective of this study is to present precise proofs of the Collatz conjecture and introduce some interesting behavior on Kaakuma sequence. We propose a novel approach that tackles the Collatz conjecture using different techniques and angles. The Collatz Conjecture, proposed by Lothar Collatz in 1937, remains one of the most intriguing unsolved problems in mathematics. The conjecture posits that, for any positive integer, applying a series of operations will eventually lead to the number 1. Despite decades of rigorous investigation and countless computational verification, a complete proof has eluded mathematicians. In this research endeavor, we embark on a comprehensive exploration of the Collatz Conjecture, aiming to shed light on its underlying principles and ultimately establish its validity. Our investigation begins by defining the Collatz function and investigating some behavior like. transformation, selective mapping, successive division, constant growth rate of inverse tree, and more. Using and analyzing the discovered properties of Collatz sequence we can show there are contradiction. In addition to this we investigate Qodaa ratio test that validates the reality of discovered behavior of Collatz sequence and works for infinite and distinct Kaakuma sequences. Our investigation culminates in the formulation of a set of conjectures encompassing lemmas and postulates, which we rigorously prove using a combination of analytical reasoning, numerical evidence, and exhaustive case analysis. These results provide compelling evidence for the veracity of the Collatz Conjecture and contribute to our understanding of the underlying mathematical structure. This proof helps to change some researchers' views on unsolved problems and offers new perspectives on probability in infinite range. In this study, we uncover the dynamic nature of the Collatz sequence and provide a reflection and interpretation of the probabilistic proof of the Collatz Conjecture.
Collatz conjecture revisited: an elementary generalization
Acta Universitatis Sapientiae, Mathematica, 2020
Collatz conjecture states that iterating the map that takes even natural number n to n 2 {n \over 2} and odd natural number n to 3n + 1, will eventually obtain 1. In this paper a new generalization of the Collatz conjecture is analyzed and some interesting results are obtained. Since Collatz conjecture can be seen as a particular case of the generalization introduced in this articule, several more general conjectures are also presented.
Some elements for the demonstration of Collatz conjecture
2015
In this paper we show the following facts: The probability of increasing A k = P (T k (x 0) > T kβ1 (x 0)), and the probability of decreasing B k = P (T k (x 0) < T kβ1 (x 0)) in step k of a Collatz procedure initiated in x 0 β N arbitrary, they are equal for all values of k. This influences on the law that generates the numbers of a Collatz sequence so that it is forced to decrease until the unit. It is also shown that in the Collatz conjecture is false for every problem an + b such that a β₯ 5 β₯ b + 2, and its probabilistic character can not be ignored if you want to get to the definitive solution, among other interesting arguments.
A proof of the Collatz conjecture
We here provide a proof of the Collatz conjecture, also known as the (3x+1) or Syracuse conjecture. The proof is organised as follows. First, three distinct fixed points are found for the third iterate of the Collatz map, which hence organise in a period 3 orbit of the original map. These are 1, 2 and 4, the elements which define the unique attracting cycle, as hypothesised by Collatz. To carry out the calculation we write the positive integers in modulo 8, obtain a closed analytical form for the associated map and determine the transitions that yield contracting or expanding iterates. Then, in the second part of the proof, we consider a probabilistic analogue of the deterministic map, where the period-3 orbit {1, 2, 4} acts as an absorbing sink. Working in this setting, we demonstrate that the quasi-stationary state sampled by the system while seeking for its deputed equilibrium is contracting and that each trajectory is bound to converge to the fixed points, as identified above. This implies in turn that all positive integer numbers converge to the periodβ3 orbit formed by the numbers {1, 2, 4}, under repeated application of the Collatz map. This proves in turn the conjecture.
Novel Theorems and Algorithms Relating to the Collatz Conjecture
International Journal of Mathematics and Mathematical Sciences
Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and algorithms that explore relationships and properties between the natural numbers, their peak values, and the conjecture. These contributions primarily analyze the number of Collatz iterations it takes for a given integer to reach 1 or a number less than itself, or the relationship between a starting number and its peak value.
A Simple binary analysis of the Collatz Conjecture
This paper presents an attempt to address the Collatz conjecture using binary representation analysis. The analysis relies on basic properties of binary numbers and elementary algebra, suggesting that sequence convergence is guided by the intrinsic properties of binary operations. The key insight is analyzing the 3n + 1 transformation in terms of its effect on binary digits. Furthermore, it is demonstrated that this approach can be generalized to similar problems by changing the base of the representation. This work explores the Collatz conjecture through a novel lens: binary representation and elementary algebraic operations. By analyzing the transformations of binary sequences, this approach highlights structural properties and recurring patterns that contribute to a deeper understanding of the conjecture's dynamics.
Base-X Conjecture and Collatz Conjecture Proof
Applied Mathematics, 2025
A new Base-X Conjecture was introduced in this paper, and Collatz Conjecture is just one case of Base-X Conjecture-Base-3 (Ternary). Based on Base-X number system property and Collatz Conjecture iteration, it has been proved that for any positive integer D, there are n and m which exist for 2 m n n D Y + =. n n D Y + is just the result built up by collecting divided by 2 of Collatz Conjecture iteration. Divided by 2 m will make the Collatz Conjecture get a result of 1 for any positive integer. Also, the Collatz Tree showed that for any odd positive number, there is only one route existing in the Collatz Tree down to 1 on Collatz Conjecture iteration.
A new conjecture equivalent to Collatz conjecture
arXiv (Cornell University), 2023
In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz conjecture.