Two Paths to Convergence: Observations of the Collatz Conjecture

Collatz conjecture, a longstanding problem in number theory, has intrigued mathematicians due to its simple formulation yet complex behavior. In this study, Moussa Model of Positive Integers is introduced, a novel mathematical framework that extends and generalizes the Collatz conjecture. This model classifies integers into distinct sets, including decaying sequences, keys, subkeys, and their multiples, each following specific transformation rules. Moussa conjecture is introduced, which postulates that an infinite number of possible functions—each with distinct formulations and variations—converge to 1, thereby providing a broader perspective on integer dynamics. To validate this model, a computational approach using Python was employed, testing the framework across a defined numerical range. The results confirm that all tested integers adhere to Moussa model, reinforcing its structural validity. Additionally, multiple variations of transformation rules are explored, demonstrating that different formulations consistently align with the proposed model. A significant outcome of this study is that if Moussa conjecture is proven, the Collatz conjecture would be directly established as a special case within this generalized framework. This study suggests potential applications of Moussa model in cryptography, steganography, and encryption systems, particularly through the representation of keys as infinite structured sets. By approaching number theory through this innovative perspective, new mathematical problems may be addressed, and additional research directions may emerge.

Number theory is the part of mathematics concerned with the mysterious and hidden properties of the integers and rational numbers (by a rational number, we mean the ratio of two integers). The congruent number problem, the written history of which can be traced back at least a millennium, is the oldest unsolved major problem in number theory, and perhaps in the whole of mathematics. We say that a right-angled triangle is “rational” if all its sides have rational length. A positive integer N is said to be “congruent” if it is the area of a rational right-angled triangle. If we multiply any congruent number N by the square of an integer, we again get a congruent number, and so it suffices to consider only those integers N that are square-free (meaning not divisible by the square of an integer >1). The congruent number problem is simply the question of deciding which square-free positive integers are, or are not, congruent numbers. Long ago, it was realized that an integer N ≥ 1 is congruent if and only if there exists a point (x, y) on the elliptic curve y2 = x3 − N2x, with rational coordinates x, y and with y ≠ 0. Until the 17th century, mathematicians made numerical tables of congruent numbers by using ingenuity to write down the corresponding rational right-angled triangles. For example, the integers 5, 6, and 7 were all known to be congruent, as they are the areas of the right-angled triangles, whose sides lengths are given respectively by [40/6, 9/6, 41/6], [3, 4, 5], and [288/60, 175/60, 337/60]. The first important theoretical result about congruent numbers was established by Fermat, who proved in the 17th century that 1 is not a congruent number. As explained in more …

It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all (odd) natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number 1, as conjectured by Lothar Collatz.

We present a rigorous proof of the Collatz conjecture through a novel analysis of bounded sequences and cycle properties. The proof establishes strict bounds on sequence behavior and demonstrates the uniqueness of the fundamental cycle, proving that all positive integers must eventually reach 1 under the Collatz iteration. Our approach combines classical techniques from number theory with careful analysis of sequence bounds to resolve this long-standing conjecture. The methodology introduces several novel techniques that may prove valuable for analyzing other iterative systems and number-theoretic conjectures.

We present a rigorous proof of the Collatz conjecture through a novel analysis of bounded sequences and cycle properties. The proof establishes strict bounds on sequence behavior and demonstrates the uniqueness of the fundamental cycle, proving that all positive integers must eventually reach 1 under the Collatz iteration. Our approach combines classical techniques from number theory with careful analysis of sequence bounds to resolve this long-standing conjecture. The methodology introduces several novel techniques that may prove valuable for analyzing other iterative systems and number-theoretic conjectures.

This paper contains a summary table of the author’s computation of the Gaussian primes of the form a + i a + i . For the values x = 1000 , 10000 , 100000 , 180000 x = 1000, 10000, 100000, 180000 , and 500000 (500000) 14000000, the following values are tabulated: G ( x ) G(x) , the numbers of Gaussisn primes a + i a + i with a ≦ 14000000 ; π 1 ( x ) a \leqq 14000000;{\pi _1}(x) , the number of primes ≦ x \leqq x congruent to 1 m o d 4 1 mod 4 ; π 3 ( x ) {\pi _3}(x) , the number of primes ≦ x \leqq x congruent to 3 m o d 4 3 mod 4 ; and G ( x ) / π 3 ( x ) G(x)/{\pi _3}(x) .

The Collatz conjecture (or 3n+1 problem) has been explored for about 86 years. In this article, we prove the Collatz conjecture. We will show that this conjecture holds for all positive integers by applying the Collatz inverse operation to the numbers that satisfy the rules of the Collatz conjecture. Finally, we will prove that there are no positive integers that do not satisfy this conjecture. Mathematics Subject Classification (2020): 11A25, 11A51, 11B50, 11B75. Keywords: Collatz operation, Collatz inverse operation, Collatz numbers.

Dynamic software watermarking is one of the major countermeasures against software licensing violations. However, conventional dynamic watermarking approaches have exhibited a number of weaknesses including exploitable payload semantics, exploitable embedding/recognition procedures, and weak correlation between payload and subject software. This paper presents a novel dynamic watermarking method, Xmark, which leverages a well-known unsolved mathematical problem referred to as the Collatz conjecture. Our method works by transforming selected conditional constructs (which originally belonged to the software to be watermarked) with a control flow obfuscation technique based on Collatz conjecture. These obfuscation routines are built in a particular way such that they are able to express a watermark in the form of iteratively executed branching activities occurred during computing the aforementioned conjecture. Exploiting the one-to-one correspondence between natural numbers and their orbits computed by the conjecture (also known as the “Hailstone sequences”), Xmark's watermark-related activities are designed to be insignificant without the pre-defined secret input. Meanwhile, being integrated with obfuscation techniques, our method is able to resist attacks based on various reverse engineering techniques on both syntax and semantic levels. Analyses and simulations indicated that Xmark could evade detections via pattern matching and model checking, and meanwhile effectively prohibit dynamic symbolic execution. We have also shown that our method could remain robust even if a watermarked software is compromised via re-obfuscation using approaches like control flow flattening.

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

With the availability of microcomputers, mathematics classes have the opportunity to explore topics in number theory that have previously been ignored or treated superficially. The exploration of problems in number theory permits a further understanding of the BASIC language by introducing new computer functions. One topic that couples number theory to programming is palindromes.

Prime numbers are an important research area with application in computer security. The security of certain cryptographic schemes critical in information security lies in the difficulty of factoring large numbers into their prime factors. Determination of the distribution of prime numbers is an important unsolved problem in number theory. Biomolecules such as DNA have extended the domains of what is considered a standard computer model. This report investigated the distribution of prime numbers generated from viral DNA. The digits of Euler’s number and Pi were found to be encoded between Bacteriophage T4 Watson-Crick DNA segments comprised of prime numbers. Key words: Prime number, number theory, molecular computing, phage T4.

We denote by A_{3.n}, the number of the form 3^{2^n}+2, where n is an integer \geq0; in other words, for every integer n\geq0,A_{3.n}=3^{2^n}+2 [it is easy to check that A_{3.0}=5,A_{3.1}=11,A_{3.2}=83, A_{3.3}=6563 and A_{3.4}=43046723; moreover, for every j\in{0,1,2,3},A_{3.j} is prime and A_{3.4} is composite ( since A_{3.4}\equiv0mod[19] )]. In this paper, we show [via elementary arithmetic congruences] the following result (T.). For every positive integer n such that n\equiv4mod[6], we have A_{3.n}-2\equiv17mod[19]; and for every integer n\geq2, we have A_{3.n}-2\equiv1mod[5]. Result (T.) immediately implies that there are infinitely many composite numbers of the form A_{3.n}. That being so, using result (T.) coupled with a special case of a Theorem of Dirichlet on arithmetic progressions, we explain why it is natural to conjecture that there are infinitely many prime numbers of the form A_{3.n}. A Fermat number is a number of the form F_n=2^{2^n}+1, where n is an integer \geq0. A Fermat composite (see Dickson (1952) or Hardy, & Wright, (1979) or Annouk, (2012)) is a non prime Fermat number and a Fermat prime is a prime Fermat number. Fermat composites and Fermat primes are characterized via divisibility in Annouk, (2012) and in Annouk, (2014). It is known (see [4] ) that for every j\in{0,1,2,3,4},F_j is a Fermat prime \left(F_0=2^{2^0}+1=3\right. and 3 is prime, F_1=2^{2^1}+1=5 and 5 is prime, F_3=2^{2^3}+1=257 and 257 is prime, and F_4=2^{2^4}+1=65537 and 65537 is prime), and it is also known (see Hardy, & Wright (1979) or Hoffman (2000)or Křížek, et al. (2002) or Keller, (2020) or Euler, (1738)or Scharlau, & Opolka, (1985) that F_5 and F_6 are Fermat composites \left(F_5=2^{2^5}+1=641\times6700417\right., and since 2013, it is known that F_{2747497}=2^{2^{2747497}}+1 is Fermat composite number). Fermat numbers have importance and their application to other sciences such as cryptography, neural networks, electronic computer, polygons with straightedge, filtering, autocorrelation, and related areas with conventional computing are clearly known. Factorization of Fermat numbers (specially for big one) is a very hard problem in number theory and cryptography too. The biggest known Fermat prime is F_4=2^{2^4}+1=65537, and this number is used in cryptography because of the fact that this prime helps the cryptography less vulnerable to the public exponent attack lowly (as mentioned "Coppersmith's short pad attack" in the literature). Practical and efficient methods are still in need for factorization of such numbers, even there are many algorithms to factorize some of such composite numbers. That being so, in this paper, we show [via elementary arithmetic congruences] the following result (E.). For every integer n>0 such that n\equiv1mod[2], we have F_n-1\equiv4mod[7]; and for every integer n\geq2, we have F_n-1\equiv1mod[j], where j\in{3,5}. Result (E.) immediately implies that there are infinitely many composite numbers of the form 2+F_n. Result (E.) also implies that the only prime of the form 4+F_n is 7 and the only primes of the form 8+F_n are twin primes 11 and 13. That being said, using result (E.) and a special case of a Theorem of Dirichlet, we conjecture that there are infinitely many primes of the form 2+F_n.

Cryptographic primitives have been constructed on a wide range of mathematical basis. For asymmetric cryptography, those bases are problems in number theory. Lattice based mathematical problems provide a basis for cryptographic algorithms. Lattice-based cryptographic algorithms are believed to be resistant to quantum computing attacks. This makes research concerning these algorithms important. What is lacking in the current literature is a comparative study of existing cryptanalysis data for these algorithms. That data can be used to guide further research into lattice-based asymmetric cryptographic primitives. This current paper provides such a comparison.