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Register a new userVerification of Collatz Conjecture: An algorithmic approach based on binary representation of integers
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Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the Collatz conjecture based on bit representation of integers. The scheme neither encounters any cycles in the so called Collatz sequence and nor the sequence grows indefinitely. Experimental results show that the Collatz sequence starting at the given integer , oscillates for finite number of times, never exceeds 1.7 times (scaling factor) size of the starting integer and finally reaches the value 1. The experimental results show strong evidence that conjecture is correct and paves a way for theoretical proof.
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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.
Given a negative $D>-(log X)^{log 2-delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of discriminant $D$. We also give an analogous upper bound for square free integers of the form $q+a<X$ where $q$ is prime and $ainmathbb Z$ is fixed. Combined with the 1/2-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers $q+a<X$ represented by a binary quadratic form of discriminant $D$, where $D$ is allowed to grow with $X$ as above. An immediate consequence of this, coming from recent work of the authors in [BF], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak proof of the conjecture by using induction with respect to the degree of the polynomials.
It is well known that the Collatz Conjecture can be reinterpreted as the Collatz Graph with root vertex 1, asking whether all positive integers are within the tree generated. It is further known that any cycle in the Collatz Graph can be represented as a tuple, given that inputting them into a function outputs an odd positive integer; yet, it is an open question as to whether there exist any tuples not of the form $(2,2,...,2)$, thus disproving the Collatz Conjecture. In this paper, we explore a variant of the Collatz Graph, which allows the 3x+1 operation to be applied to both even and odd integers. We prove an analogous function for this variant, called the Loosened Collatz Function (LCF), and observe various properties of the LCF in relation to tuples and outputs. We then analyse data on the numbers that are in cycles and the length of tuples that represent circuits. We prove a certain underlying unique factorisation monoid structure for tuples to the LCF and provide a geometric interpretation of satisfying tuples in higher dimensions. Research into this variant of the Collatz Graph may provide reason as to why there exist no cycles in the Collatz Graph.
We present a formulation of the Collatz conjecture that is potentially more amenable to modeling and analysis by automated termination checking tools.
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