Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Glide of 3x+1 Problem
92
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
For any positive integer $n$, define an iterated function $$ f(n)=left{begin{array}{ll} n/2, & mbox{$n$ even,} 3n+1, & mbox{$n$ odd.} end{array} right. $$ Suppose $k$ (if it exists) is the lowest number such that $f^{k}(n)<n$, and there are $O(n)$ multiply by three and add one and $E(n)$ divide by two from $n$ to $f^{k}(n)$, then there must be $$ 2^{E(n)-1}<3^{O(n)}<2^{E(n)}. $$ Our results confirm the conjecture proposed by Terras in 1976.
rate research
Read More
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.
Let $e(s)$ be the error term of the hyperbolic circle problem, and denote by $e_alpha(s)$ the fractional integral to order $alpha$ of $e(s)$. We prove that for any small $alpha>0$ the asymptotic variance of $e_alpha(s)$ is finite, and given by an explicit expression. Moreover, we prove that $e_alpha(s)$ has a limiting distribution.
In this note, I study a comparison map between a motivic and {e}tale cohomology group of an elliptic curve over $mathbb{Q}$ just outside the range of Voevodskys isomorphism theorem. I show that the property of an appropriate version of the map being an isomorphism is equivalent to certain arithmetical properties of the elliptic curve.
For g,n coprime integers, let l_g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l_g(n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l_g(p) as p <= x ranges over primes.
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harmans sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selbergs eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
