ARTICLE | doi:10.20944/preprints202309.1200.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: binary representation; composite function; Collatz conjecture; ultimately periodic sequence; chaos
Online: 19 September 2023 (08:07:22 CEST)
The two main strategies used in this study are the binary representation and the decomposition of a natural number into many compound functions of odd function and even function. The Collatz conjecture regarding odd even numbers in number theory can be examined and discussed using them in this way: The sequence created by the finite iterations of the Collatz function becomes the ultimately periodic sequence if any natural number is the beginning value, proving the conjecture that has been held for 85 years.
ARTICLE | doi:10.20944/preprints202304.0931.v2
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Collatz conjecture; binary number; ultimately periodic; chaos
Online: 19 May 2023 (16:16:49 CEST)
The following is the Collatz Conjecture: Suppose we begin with a positive number, multiply it by 3 and add 1 if it is odd, and divide it by 2 if it is even. Then continue doing this as long as you can. Will it matter where you start, whether you end up at the number 1? The Collatz conjecture has been studied for around 85 years. We transform the Collatz function from decimal to binary, then use the binary string's character to prove the Collatz conjecture. In addition, we use mathematics to give another interpretation to chaos, which is the ultimately periodic positive integer sequence.
ARTICLE | doi:10.20944/preprints202301.0163.v5
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: The Collatz conjecture
Online: 20 February 2023 (10:26:07 CET)
The 3x+1 problem asks the following: Suppose we start with a positive integer, and if it is odd then multiply it by 3 and add 1, and if it is even, divide it by 2. Then repeat this process as long as you can. Do you eventually reach the integer 1, no matter what you started with? Collatz conjecture (or 3n+1 problem) has been explored for about 85 years. In this paper, we convert an integer number from decimal to binary number, and convert the Collatz function to binary function, which is multiplication and division of two binary numbers. Finally the iternation of the Collatz function, eventually reach the integer 1, thus we solve the 3x+1 problem completely.
ARTICLE | doi:10.20944/preprints202008.0272.v1
Subject: Computer Science And Mathematics, Computational Mathematics Keywords: Bohemian; Toeplitz matrix; Hessenberg matrix; tridiagonal matrix; pentadiagonal matrix
Online: 12 August 2020 (06:00:31 CEST)
In this paper, we deduce explicit formulas to evaluate the determinants of nonsymmetrical structure Toeplitz Bohemians by two determinants of specific Hessenberg Toeplitz matrices, which are linear combinations in terms of determinants of specific Hessenberg Toeplitz matrices. We get some new results very di¤erent from [Massimiliano Fasi, Gian Maria Negri Porzio, Determinants of normalized upper Hessenberg matrices, Electronic Journal of Linear Algebra, Volume 36, pp. 352-366, June 2020].