Orbits Theory. A Complete Proof of the Collatz Conjecture

In this work a complete proof of the Collatz Conjecture is presented. The solution assumes as hypothesis that Collatz's Conjecture is a consequence. We found that every natural number n_i∈N can be calculated starting from 1, using the function n_i=((2^(i-Ω)-C))⁄3^Ω , where: i≥0 represents the number of steps (operations of multiplications by two subtractions of one and divisions by three) needed to get from 1 to n_i, Ω≥0 represents the number of multiplications by three required and 0≤C≤2^(i-⌊i/3⌋ )-2^((i mod 3)) 3^⌊i/3⌋ is an accumulative constant that takes into account the order in which the operations of multiplication and division have been performed. Reversing the inversion, we have obtained the function: ((3^Ω n_i+C))⁄2^(i-Ω)=1 that proves that Collatz Conjecture it’s a consequence of the above and also proofs that Collatz Conjecture it’s true since ((3^Ω n_i+C))⁄2^(i-Ω) is the recursive form of the Collatz’s function.

Orbits Theory. A Complete Proof of the Collatz Conjecture

In this work a complete proof of the Collatz Conjecture is presented. The solution assumes as hypothesis that Collatz's Conjecture is a consequence. We found that every natural number n_i∈N can be calculated starting from 1, using the function n_i=((2^(i-Ω)-C))⁄3^Ω , where: i≥0 represents the number of steps (operations of multiplications by two subtractions of one and divisions by three) needed to get from 1 to n_i, Ω≥0 represents the number of multiplications by three required and 0≤C≤2^(i-⌊i/3⌋ )-2^((i mod 3)) 3^⌊i/3⌋ is an accumulative constant that takes into account the order in which the operations of multiplication and division have been performed. Reversing the inversion, we have obtained the function: ((3^Ω n_i+C))⁄2^(i-Ω)=1 that proves that Collatz Conjecture it’s a consequence of the above and also proofs that Collatz Conjecture it’s true since ((3^Ω n_i+C))⁄2^(i-Ω) is the recursive form of the Collatz’s function.

Collatz conjecture(3X+1)There is a hidden theorem ω1: If α Establishment, necessity (3α+ 1) Established

From a number theory “Collatz conjecture (3X+1)”, Human beings use a large amount of computer data, so far no counterexample has been found. Does mathematical logic support " Collatz conjecture (3X+1)? Collatz conjecture (3X+1) There is a hidden theorem ω1 : If x holds, it must be (3X+1). In reality, human beings will only (3X+1) deduce that x holds.Example: an integer a, and a = 3b +1, b∈N. If b→3x +1 holds. There must be: a→3x +1 is established.In this way, there is no need to deduce (3b + 1).

Collatz conjecture(3X+1)There is a hidden theorem ω1: If α Establishment, necessity (3α+ 1) Established

From a number theory “Collatz conjecture (3X+1)”, Human beings use a large amount of computer data, so far no counterexample has been found. Does mathematical logic support " Collatz conjecture (3X+1)? Collatz conjecture (3X+1) There is a hidden theorem ω1 : If x holds, it must be (3X+1). In reality, human beings will only (3X+1) deduce that x holds.Example: an integer a, and a = 3b +1, b∈N. If b→3x +1 holds. There must be: a→3x +1 is established.In this way, there is no need to deduce (3b + 1).

Collatz conjecture(3X+1)There is a hidden theorem ω1: If α Establishment, necessity (3α+ 1) Established

From a number theory “Collatz conjecture (3X+1)”, Human beings use a large amount of computer data, so far no counterexample has been found. Does mathematical logic support " Collatz conjecture (3X+1)? Collatz conjecture (3X+1) There is a hidden theorem ω1 : If x holds, it must be (3X+1). In reality, human beings will only (3X+1) deduce that x holds.Example: an integer a, and a = 3b +1, b∈N. If b→3x +1 holds. There must be: a→3x +1 is established.In this way, there is no need to deduce (3b + 1).

Novel Theorems and Algorithms Relating to the Collatz Conjecture

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.

The word problem for one-relation monoids: a survey

This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem.