Related papers: Collatz Conjecture: Patterns Within
In this article, we give two different proofs of why the Collatz Conjecture is false.
We introduce the \emph{Collatz conjecture} and its history. Some definition that this conjecture has, will be expressed and with these we try to explain some good lemma to justify the main properties of the \emph{Collatz conjecture}. With…
Information and individual activities often spread globally through the network of social ties. While social contagion phenomena have been extensively studied within the framework of threshold models, it is common to make an assumption that…
Condensation is the phenomenon whereby one of a sum of random variables contributes a finite fraction to the sum. It is manifested as an aggregation phenomenon in diverse physical systems such as coalescence in granular media, jamming in…
A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring…
We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…
In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new…
In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…
We consider a version of the Watts cascade model on directed multiplex configuration model networks, and present a detailed analysis of the cascade size, single-seed cascade probability and cascade condition. We then introduce a smaller…
We study sequences of random numbers {Z[1],Z[2],Z[3],...,Z[n]} -- which can be considered random walks with reflecting barriers -- and define their "types" according to whether Z[i] > Z[i+1], (a down-movement), or Z[i] < Z[i+1]…
The cluster structures that can be observed in the first few level sets of the Collatz tree are maintained through all its levels, provided that the orbit steadiness \[ \prod_{\substack{k \in R(n)\\ k \equiv 4\ (\mathrm{mod}\ 6)}}…
The Collatz conjecture (also known as the $3x+1$ problem) concerns the behavior of the discrete dynamical system on the positive integers defined by iteration of the so-called $3x + 1$ function. We investigate analogous dynamical systems in…
Computational properties of networks that can undergo cascades are examined. It is shown that universal Boolean logic circuits can be computed by a global cascade having antagonistic interactions. Determinism and cascade frequency of this…
We present a formulation of the Collatz conjecture that is potentially more amenable to modeling and analysis by automated termination checking tools.
The 3n+1, or Collatz problem, is one of the hardest math problems, yet still unsolved. The Collatz conjecture is to prove or disprove that the Collatz sequences COL(n) always eventually reach the number of 1, for all n belongs to N+ (all…
Structured prediction tasks pose a fundamental trade-off between the need for model complexity to increase predictive power and the limited computational resources for inference in the exponentially-sized output spaces such models require.…
We present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of random networks with arbitrary degree distribution and nonzero clustering introduced in [M.E.J. Newman, Phys.…
The Collatz variations pattern seems not to have any recurrence relation between numbers. But knowing that there is at least a natural number that converges after several iterations we construct a function $f_{X,Y}$ that is equal to the…
A source sequence is to be guessed with some fidelity based on a rate-limited description of an observed sequence with which it is correlated. The trade-off between the description rate and the exponential growth rate of the least power…
Three events in a probability space form a conjunctive fork if they satisfy specific constraints on conditional independence and covariances. Patterns of conjunctive forks within collections of events are characterized by means of systems…