scholarly journals Collatz Dynamics is Partitioned by Residue Class Regularly

2020 ◽  
Author(s):  
Wei Ren
Keyword(s):  
Positive Integer ◽  
Residue Class ◽  
Collatz Conjecture ◽  
If Current ◽  
Positive Integers

<div>We propose Reduced Collatz Conjecture that is equivalent to Collatz</div><div>Conjecture, which states that every positive integer can return to</div><div>an integer less than it, instead of 1. Reduced Collatz Conjecture</div><div>should be easier because some properties are presented in reduced</div><div>dynamics, rather than in original dynamics (e.g., ratio and period).</div><div>Reduced dynamics is a computation sequence from starting integer to</div><div>the first integer less than it, and original dynamics is a</div><div>computation sequence from starting integer to 1. Reduced dynamics is</div><div>a component of original dynamics. We denote dynamics of x as a</div><div>sequence of either computations in terms of ``I'' that represents</div><div>(3*x+1)/2 and ``O'' that represents x/2. Here 3*x+1 and x/2 are</div><div>combined together, because 3*x+1 is always even and followed by x/2.</div><div>We formally prove that all positive integers are partitioned into</div><div>two halves and either presents ``I'' or ``O'' in next ongoing</div><div>computation. More specifically, (1) if any positive integer x that</div><div>is i module $2^t$ (i is an odd integer) is given, then the first t</div><div>computations (each one is either ``I'' or ``O'' corresponding to</div><div>whether current integer is odd or even) will be identical with that</div><div>of i. (2) If current integer after t computations (in terms of ``I''</div><div>or ``O'') is less than x, then reduced dynamics of x is available.</div><div>Otherwise, the residue class of x (namely, i module $2^t$) can be</div><div>partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$</div><div>module $2^{t+1}$), and either half presents ``I'' or ``O'' in</div><div>intermediately forthcoming (t+1)-th computation.</div>

scholarly journals Collatz Dynamics is Partitioned by Residue Class Regularly

2020 ◽  
Author(s):  
Wei Ren
Keyword(s):  
Positive Integer ◽  
Residue Class ◽  
Collatz Conjecture ◽  
If Current ◽  
Positive Integers

<div>We propose Reduced Collatz Conjecture that is equivalent to Collatz</div><div>Conjecture, which states that every positive integer can return to</div><div>an integer less than it, instead of 1. Reduced Collatz Conjecture</div><div>should be easier because some properties are presented in reduced</div><div>dynamics, rather than in original dynamics (e.g., ratio and period).</div><div>Reduced dynamics is a computation sequence from starting integer to</div><div>the first integer less than it, and original dynamics is a</div><div>computation sequence from starting integer to 1. Reduced dynamics is</div><div>a component of original dynamics. We denote dynamics of x as a</div><div>sequence of either computations in terms of ``I'' that represents</div><div>(3*x+1)/2 and ``O'' that represents x/2. Here 3*x+1 and x/2 are</div><div>combined together, because 3*x+1 is always even and followed by x/2.</div><div>We formally prove that all positive integers are partitioned into</div><div>two halves and either presents ``I'' or ``O'' in next ongoing</div><div>computation. More specifically, (1) if any positive integer x that</div><div>is i module $2^t$ (i is an odd integer) is given, then the first t</div><div>computations (each one is either ``I'' or ``O'' corresponding to</div><div>whether current integer is odd or even) will be identical with that</div><div>of i. (2) If current integer after t computations (in terms of ``I''</div><div>or ``O'') is less than x, then reduced dynamics of x is available.</div><div>Otherwise, the residue class of x (namely, i module $2^t$) can be</div><div>partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$</div><div>module $2^{t+1}$), and either half presents ``I'' or ``O'' in</div><div>intermediately forthcoming (t+1)-th computation.</div>

PARTITIONS WITH PARTS IN A FINITE SET

2006 ◽  
Vol 02 (03) ◽  
pp. 455-468 ◽  
Author(s):  
ØYSTEIN J. RØDSETH ◽  
JAMES A. SELLERS
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Residue Class ◽  
Finite Set ◽  
Positive Integers

For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement.

scholarly journals Reduced Collatz Dynamics Data Reveals Properties for the Future Proof of Collatz Conjecture

Data ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 89 ◽  
Author(s):  
Ren
Keyword(s):  
Upper Bound ◽  
Residue Class ◽  
Data Set ◽  
Source Codes ◽  
Collatz Conjecture ◽  
The Future ◽  
One To One ◽  
Positive Integers ◽  
Minimal Integer

Collatz conjecture is also known as 3X + 1 conjecture. For verifying the conjecture, we designed an algorithm that can output reduced dynamics (occurred 3 × x+1 or x/2 computations from a starting integer to the first integer smaller than the starting integer) and original dynamics of integers (from a starting integer to 1). Especially, the starting integer has no upper bound. That is, extremely large integers with length of about 100,000 bits, e.g., 2100000 − 1, can be verified for Collatz conjecture, which is much larger than current upper bound (about 260). We analyze the properties of those data (e.g., reduced dynamics) and discover the following laws; reduced dynamics is periodic and the period is the length of its reduced dynamics; the count of x/2 equals to minimal integer that is not less than the count of (3 × x + 1)/2 times ln(1.5)/ln(2). Besides, we observe that all integers are partitioned regularly in half and half iteratively along with the prolonging of reduced dynamics, thus given a reduced dynamics we can compute a residue class that presents this reduced dynamics by a proposed algorithm. It creates one-to-one mapping between a reduced dynamics and a residue class. These observations from data can reveal the properties of reduced dynamics, which are proved mathematically in our other papers (see references). If it can be proved that every integer has reduced dynamics, then every integer will have original dynamics (i.e., Collatz conjecture will be true). The data set includes reduced dynamics of all odd positive integers in [3, 99999999] whose remainder is 3 when dividing 4, original dynamics of some extremely large integers, and all computer source codes in C that implement our proposed algorithms for generating data (i.e., reduced or original dynamics).

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Congruences involving alternating harmonic sums modulo pα qβ

2018 ◽  
Vol 68 (5) ◽  
pp. 975-980
Author(s):  
Zhongyan Shen ◽  
Tianxin Cai
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Mod P

Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

scholarly journals Partitioning the positive integers with higher order recurrences

1991 ◽  
Vol 14 (3) ◽  
pp. 457-462 ◽  
Author(s):  
Clark Kimberling
Keyword(s):  
Positive Integer ◽  
Positive Root ◽  
Irrational Number ◽  
Higher Order ◽  
Typical Result ◽  
Algebraic Integers ◽  
Positive Integers

Associated with any irrational numberα>1and the functiong(n)=[αn+12]is an array{s(i,j)}of positive integers defined inductively as follows:s(1,1)=1,s(1,j)=g(s(1,j−1))for allj≥2,s(i,1)=the least positive integer not amongs(h,j)forh≤i−1fori≥2, ands(i,j)=g(s(i,j−1))forj≥2. This work considers algebraic integersαof degree≥3for which the rows of the arrays(i,j)partition the set of positive integers. Such an array is called a Stolarsky array. A typical result is the following (Corollary 2): ifαis the positive root ofxk−xk−1−…−x−1fork≥3, thens(i,j)is a Stolarsky array.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

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