Related papers: Statistical (3x+1) -- problem
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…
We formulate conditions for convergence of Laws of Large Numbers and show its links with of the parts of mathematical analysis such as summation theory, convergence of orthogonal series. We present also applications of the Law of Large…
A perfect number is a number whose divisors add up to twice the number itself. The existence of odd perfect numbers is a millennia-old unsolved problem. This note proposes a proof of the nonexistence of odd perfect numbers. More generally,…
We consider a generalization of the Thirring model in 2+1 dimensions at finite density. We employ stochastic quantization and check for the applicability in the finite density case to circumvent the sign problem. To this end we derive…
The main features of the statistical approach to inverse problems are described on the example of a linear model with additive noise. The approach does not use any Bayesian hypothesis regarding an unknown object; instead, the standard…
We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the…
We consider a generalized Thirring model in 0+1 dimensions at finite density. In order to deal with the resulting sign problem we employ stochastic quantization, i.e., a complex Langevin evolution. We investigate the convergence properties…
The $3x+1$ problem, also called the Collatz conjecture, is a very interesting unsolved mathematical problem related to computer science. This paper generalized this problem by relaxing the constraints, i.e., generalizing this deterministic…
A recast of the standard residue-class analysis of the 3x+1 (Collatz) map in terms of two elementary operators on arithmetic progressions. The resulting calculus (i) splits any progression into its even and odd subsequences in a single…
Numerical relativity is finally approaching a state where the evolution of rather general (3+1)-dimensional data sets can be computed in order to solve the Einstein equations. After a general introduction, three topics of current interest…
We give an historical account, including recent progress, on some problems of Erd\H os in number theory.
The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific…
A "truncation" of Pascal's triangle is a triangular array of numbers that satisfies the usual Pascal recurrence but with a boundary condition that declares some terminal set of numbers along each row of the array to be zero. Presented here…
In this paper we study the equation $$ x^k + (x+1)^k = y^n,\quad n\geq 3, $$ when $k\equiv 2\pmod{4}$. We prove that the only solutions are for $x=0, -1$ when $6\leq k\leq 100$ or for a $k$ with odd prime factors congruent to $3\pmod{4}$.…
We describe a statistical hypothesis test for the presence of a signal. The test allows the researcher to fix the signal location and/or width a priori, or perform a search to find the signal region that maximizes the signal. The background…
We present the conclusive mathematical structure behind Tsallis statistics. We obtain mainly the following five theoretical results: (i) the one-to-one correspondence between the q-multinomial coefficient and Tsallis entropy, (ii) symmetry…
The Collatz conjecture asserts that repeatedly iterating $f(x) = (3x + 1)/2^{a(x)}$, where $a(x)$ is the highest exponent for which $2^{a(x)}$ exactly divides $3x+1$, always lead to $1$ for any odd positive integer $x$. Here, we present an…
Let k>2 be a fixed integer exponent and let \theta > 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^{\theta}N^{1/10}) ways, providing…
Sequence of numbers generated by the recurrence relation based on the Collatz conjecture is investigated. An arithmetic operation on the Collatz conjecture is called descending operation, and ascending operation is carried out reversely to…
This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szab\'{o} Theorem…