Decomposition into weight * level + jump and application to a new classification of primes

Abstract: In this paper we introduce an Euclidean decomposition of elements a_n of an increasing sequence of natural numbers into weight * level + jump which we use to classify the numbers a_n either by weight or by level. We then show that this decomposition can be seen as a generalization of the sieve of Eratosthenes (which is the particular case of the whole sequence of natural numbers). We apply this decomposition to prime numbers in order to obtain a new classification of primes, we analyze a few properties of this classification and we make a series of conjectures based on numerical data. Finally we show how composite numbers and 2-almost primes behave under the decomposition.

• The paper presents a novel decomposition algorithm that expresses natural numbers as weight × level + jump to distinguish between prime and composite numbers.
• It generalizes the sieve of Eratosthenes by leveraging numerical patterns to reveal structural properties in primes and composites.
• The work offers insights into prime classification with potential applications in factorization and cryptographic methods.

Decomposition into weight × level + jump and its Implications on Prime Classification

The paper proposes a novel method for decomposing elements of an increasing sequence of natural numbers into a format defined as weight × level + jump. This decomposition serves not only as an analytical tool for number classification but also generalizes the classical sieve of Eratosthenes. By implementing this framework, the paper argues for a new classification of prime numbers and explores the properties and implications of this classification.

Key Contributions

1. Decomposition Algorithm: The core of the paper introduces an algorithm that takes an increasing sequence of positive integers as input. Each element $a_n$ in the sequence is then expressed as a unique triplet $(k_n, L_n, d_n)$, where $k_n$ is the weight, $L_n$ is the level, and $d_n$ is the jump. The paper defines these concepts with mathematical rigor and demonstrates the conditions necessary for the decomposition to be valid.
2. Generalized Sieve of Eratosthenes: The paper presents the decomposition method as a generalization of the classical sieve of Eratosthenes. By applying the decomposition to natural numbers, it reveals a parallel with the sieve, highlighting that numbers decomposable into a lower level signify prime numbers, whereas those characterized by higher weight are indicative of composite numbers.
3. Classification of Primes: By applying the decomposition to prime numbers, the algorithm yields a distinct classification based on the properties of weight and level. The paper details how different classes of primes can be extracted and characterized using this method, providing conjectures supported by numerical data.
4. Behavior of Composite numbers and Almost Primes: The analysis extends beyond prime numbers, considering how composite numbers and 2-almost primes (numbers with exactly two prime factors) fit into this decomposition framework, further supporting the utility of the method in broader number-theoretic inquiries.

Numerical Results and Conjectures

The paper presents an in-depth numerical analysis of sequences, verifying the algorithm's efficacy and proposing conjectures based on observed patterns within prime and composite classifications. While exact numerical results or conjectures are not described in detail here, the paper emphasizes the method's differentiation between sequences and its implications for identifying numerical properties and patterns.

Implications

Practical Implications: This decomposition provides a new lens for identifying and categorizing prime numbers efficiently, potentially offering alternative methods for computing and analyzing primes in large datasets without relying solely on traditional sieving techniques.

Theoretical Implications: The generalization of Eratosthenes' sieve presents theoretical ramifications in the study of number theory, particularly in understanding primes' structural properties. This approach can inspire further theoretical exploration into number decomposition and its applications to different mathematical problems.

Future Developments

The paper suggests several future directions for this line of research. The evolution of the algorithm could potentially lead to improved methods for factorization, cryptographic applications, and extending the classification to higher-dimensional number sets. Moreover, further exploration and refinement could lead to proofs that substantiate the conjectures proposed.

In conclusion, the decomposition into weight × level + jump provides a robust framework for re-envisioning prime and composite number classification. It holds promise both as a computational approach and a theoretical foundation for future research in number theory and related fields.