arXiv subjects

Moustafa Ibrahim

Publications and source records attributed to Moustafa Ibrahim.

7 recordsLinked to original sources

arXiv2025

On the Emergence of the Quanta Prime Sequence

This paper presents the Quanta Prime Sequence (QPS) and its foundational theorem, showcasing a unique class of polynomials with substantial implications. The study uncovers profound connections between Quanta Prime numbers and essential sequences in number theory and cryptography. The investigation highlights the sequence's contribution to the emergence of new primes and its embodiment of core mathematical constructs, including Mersenne numbers, Fermat numbers, Lucas numbers, Fibonacci numbers, the Chebyshev sequence, and the Dickson sequence. The comprehensive analysis emphasizes the sequence's intrinsic relevance to the Lucas-Lehmer primality test. This research positions the Quanta Prime sequence as a pivotal tool in cryptographic applications, offering novel representations of critical mathematical structures. Additionally, a new result linking the Quanta Prime sequence to the Harmonic series is introduced, hinting at potential progress in understanding the Riemann Hypothesis.

math.GM↗

arXiv2024

Generalizing the Eight Levels Theorem: A Journey to Mersenne Prime Discoveries and New Polynomial Classes

Mersenne primes, renowned for their captivating form as $2^p - 1$, have intrigued mathematicians for centuries. In this paper, we embark on a captivating quest to unveil the intricate nature of Mersenne primes, seamlessly integrating methods with the Eight Levels Theorem. Initially, we extend the Eight Levels Theorem and introduce an innovative approach that harmoniously combines arithmetic and differential techniques to compute the coefficients of the polynomial expansions of $x^n + y^n$ in terms of binary quadratic forms. This endeavor leads us to the genesis of novel polynomial sequences as we scrutinize the coefficients within this expansion. Our research unearths previously uncharted connections between Mersenne numbers and the derivatives of specific polynomial sequences. By forging this linkage, we not only enhance our understanding of Mersenne primes but also bridge the divide between well-established sequences in number theory and differential equations. This broadens the applicability of our findings across diverse scientific domains, revealing fresh avenues for exploration in number theory and beyond. These polynomial bridges serve as conduits between these sequences, unveiling exciting prospects for future research. This interdisciplinary exploration opens up exciting possibilities for the broader implications of Mersenne primes, extending their significance beyond the realm of pure mathematics. In our paper, we also delve into the intriguing influence of the Golden Ratio constant, which unveils segments reminiscent of the beauty found in nature, adding an unexpected dimension to the world of arithmetic. In this harmonious journey, the allure of Mersenne primes resonates through the symphony of mathematical discovery.

math.GM↗

arXiv2023

On the Eight Levels theorem and applications towards Lucas-Lehmer primality test for Mersenne primes, I

Lucas-Lehmer test is the current standard algorithm used for testing the primality of Mersenne numbers, but it may have limitations in terms of its efficiency and accuracy. Developing new algorithms or improving upon existing ones could potentially improve the search for Mersenne primes and the understanding of the distribution of Mersenne primes and composites. The development of new versions of the primality test for Mersenne numbers could help to speed up the search for new Mersenne primes by improving the efficiency of the algorithm. This could potentially lead to the discovery of new Mersenne primes that were previously beyond the reach of current computational resources. The current paper proves what the author called the Eight Levels Theorem and then highlights and proves three new different versions for Lucas-Lehmer primality test for Mersenne primes and also gives a new criterion for Mersenne compositeness.

math.NT↗

arXiv2021

On The Emergence of a New Prime Number And Omega Sequences

This paper highlights the emergence of the Omega sequence in number theory and its connection with the emergence of a new prime number, and also highlights its theoretical applications for Lucas-Lehmer primality test, and Euclid-Euler theory for even perfect numbers. We also show that Omega sequences unify and give new representations for Mersenne numbers, Fermat numbers, Lucas numbers, Fibonacci numbers, Chebyshev sequence, Dickson sequence, and others.

math.GM↗

arXiv2021

A new explicit expansion approach to Mersenne primes

This paper first proves what the author called the Eight Levels Theorem and then highlights a new explicit expansion approach to Lucas-Lehmer primality test for Mersenne primes and gives a new criterion for Mersenne compositeness. Also, we prove four new combinatorial identities.

math.GM↗

arXiv2020

New expansions for $x^n \pm y^n$ in terms of quadratic forms

We prove new theorems for the polynomial expansions of $x^n \pm y^n$ in terms of the binary quadratic forms $αx^2 + βxy + αy^2 $ and $a x^2 + bxy + a y^2 $. The paper gives new arithmetic differential approach to compute the coefficients. Also, the paper gives generalization to well-known polynomial identity in the history of number theory. The paper highlights the emergence of a new class of polynomials that unify many well-known sequences including the Chebyshev polynomials of the first and second kind, Dickson polynomials of the first and second kind, Lucas and Fibonacci numbers, Mersenne numbers, Pell polynomials, Pell-Lucas polynomials, and Fermat numbers. Also, this paper highlights the emergence of the notions of trajectories and orbits of certain integers that passes through many well-known polynomials and sequences. The Lucas-Fibonacci trajectory, the Lucas-Pell trajectory, the Fibonacci-Pell trajectory, the Fibonacci-Lucas trajectory, the Chebyshev-Dickson trajectory of the first kind, the Chebyshev-Dickson trajectory of the second kind, and others are new trajectories included in this paper. Also, the Lucas orbit, Fibonacci orbit, Mersenne orbit, Lucas-Fibonacci orbit, Fermat orbit, and others are new orbits included in this paper.

math.GM↗

arXiv2016

A Hierarchical Deep Temporal Model for Group Activity Recognition

In group activity recognition, the temporal dynamics of the whole activity can be inferred based on the dynamics of the individual people representing the activity. We build a deep model to capture these dynamics based on LSTM (long-short term memory) models. To make use of these ob- servations, we present a 2-stage deep temporal model for the group activity recognition problem. In our model, a LSTM model is designed to represent action dynamics of in- dividual people in a sequence and another LSTM model is designed to aggregate human-level information for whole activity understanding. We evaluate our model over two datasets: the collective activity dataset and a new volley- ball dataset. Experimental results demonstrate that our proposed model improves group activity recognition perfor- mance with compared to baseline methods.

cs.CV↗