Collatz Conjecture - The Proof (original) (raw)
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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 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 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: