Abstract Generalized commutative quaternions generalize elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper, we use the Mersenne numbers and polynomials in the theory of these quaternions. We introduce and study generalized commutative Mersenne quaternion polynomials and generalized commutative Mersenne–Lucas quaternion polynomials.

The emergence of the Mersenne Star: a new geometry of prime numbers bridging physics and artificial general intelligence

It is demonstrated that neuromorphic materials designed for computational tasks can be effectively implemented by drawing an analogy with trigger-based systems built upon classical binary elements. Among the most promising approaches in this context are systems that perform computations based on the Residue Number System (RNS). A specific implementation of a trigger-based adder employing the proposed methodology is presented and tested through simulation modeling. This adder utilizes the representation of natural numbers as elements of a subtraction ring modulo P, where P is the product of Mersenne prime numbers. This configuration enables component-wise, independent execution of arithmetic operations. It is further shown that analogous trigger-based systems can be realized using recurrent neural network analogs, particularly those implemented with neuromorphic materials. The study emphasizes that it is possible to construct a neural network, especially one based on neuromorphic substrates, that can perform logical operations equivalent to those executed by conventional binary circuitry. A key challenge in the proposed approach lies in implementing an operation analogous to the carry mechanism employed in classical binary adders. An algorithm addressing this issue is proposed, based on the transition from computations modulo P to computations modulo 2P.

We offer detailed proofs of some properties of the Rule 60 cellular automaton on a ring with a Mersenne number circumference. We then use these properties to define a propagator, and demonstrate its use to construct all the ground state configurations of the classical Newman–Moore model on a square lattice of the same size. In this particular case, the number of ground states is equal to half of the available spin configurations in any given row of the lattice.

This paper investigates Tribonacci numbers can be expressed as either the sum or difference of two distinct powers of 2. Namely, we address the problem of expressing Tribonacci numbers in the form T_n=2^x±2^y in positive integers with 1≤y≤x. Our findings reveal specific instances where such representations are possible, including examples like the seventh Tribonacci number expressed both as the sum and the difference of powers of 2. Additionally, we identify Tribonacci numbers that can be represented as the differences of Mersenne numbers, specifically, the numbers 2, 4, 24, and 504. These results enhance the understanding of the structural properties of Tribonacci sequences and their relationships with exponential and Mersenne-based number systems.

Fermat or Mersenne numbers as products of two k-Pell numbers

In our study we define bicomplex (p,q)- Mersenne numbers. Utilizing these numbers, we present bicomplex (p,q)- Mersenne quaternions which are a generalization of Mersenne quaternions. All these sequences have second order recurrence relations. We obtain Binet formula, the generating functions, the exponential generating function, Catalan identity, Cassini identity, D’ocagne’s identity, summation formula for both. We also introduce matrix form of bicomplex (p,q)- Mersenne quaternions. Also, we introduce a novel generalization of Mersenne numbers by incorporating Catalan numbers into the quaternionic (p,q)- Mersenne sequence. By extending Mersenne numbers to a quaternionic framework, we establish a new connection between Catalan numbers and quaternion structures. This approach provides deeper insights into the algebraic and combinatorial properties of these generalized sequences. Potential future research directions include further exploration of their structural characteristics and applications in number theory and related fields. Also, We introduced the quaternion-type Catalan transform for Mersenne numbers, demonstrating how it amplifies growth from exponential to super-exponential and faster-than-quadratic rates. This transformation enhances the sequence’s structural complexity and information density.

In this work, we investigate the binomial transforms and Catalan transform of the $k$-Mersenne and $k$-Mersenne-Lucas numbers and examine the new integer sequences. We apply the $p$-binomial, rising $p$-binomial, and falling $p$-binomial transforms to the $k$-Mersenne sequences and present the associated generating and exponential generating functions for these transforms. Lastly, we provide the corresponding Binet-type formulas and recurrence relations for binomial transforms of the $k$-Mersenne ($k$-Mersenne-Lucas) numbers. These results are supported by numerical illustrations.

In this work we connect Bernoulli numbers and polynomials to Mersenne numbers via recurrence relations. We find two explicit formulas of Bernoulli numbers by means of Mersenne numbers different from those given by F. Qi and X. Y. Chen et al. To end with other interesting relationships, which serve as bridges between the Bernoulli polynomials and Mersenne numbers.

Mersenne prime numbers, expressed in the form (2n − 1), have long captivated researchers due to their unique properties. The presented work aims to develop a symmetric cryptographic algorithm using a novel technique based on the logical properties of Mersenne primes. Existing encryption algorithms exhibit certain challenges, such as scalability and design complexity. The proposed novel modular multiplicative inverse property over Mersenne primes simplifies the encryption/decryption process. The simplification is achieved by computing the multiplicative inverse using cyclic bit shift operation. The proposed image encryption/decryption scheme involves a series of exor, complement, bit shift, and modular multiplicative inversion operations. The image is segmented into blocks of 521 bits. Each of these blocks is encrypted using a 521-bit key, ensuring high entropy and low predictability. The inclusion of cyclic bit shifting and XOR operations in the encryption/decryption process enhances the diffusion properties and resistance against attacks. This approach was experimentally proven to secure the image data while preserving the image structure. The experimental results demonstrate significant improvements in security metrics, including key sensitivity and correlation coefficients, confirming the technique’s effectiveness against cryptographic attacks. Overall, this method offers a scalable and secure solution for encrypting high-resolution digital images without compromising computational efficiency.

In this article, we present the generalized Gaussian Mersenne numbers with arbitrary initial values and discuss two particular cases, namely, Gaussian Mersenne and Gaussian Mersenne-Lucas numbers. We present their various algebraic properties such as Binet’s formula, negatively subscripted elements, Catalans’s, Cassini’s, and d’Ocagne’s identities, partial sum, binomial sum, generating and exponential generating functions, etc. In addition, we study a new generalized sequence arising from the explicit expression made with the characteristic roots and refer to them as the k-generalized Gaussian Mersenne numbers. We present various identities of them and show their connections with the generalized Gaussian Mersenne numbers.

On the emergence of the Quanta Prime sequence

In this study, by making use of the symmetrizing operator ?k+1a1a2 we introduce a new theorem. By using this theorem we give a new class of generating functions of the products of (p, q)-Fibonacci numbers, (p, q)-Lucas numbers, (p, q)-Pell numbers, (p, q)-Pell Lucas numbers, (p, q)-Jacobsthal numbers and (p, q) -Jacobsthal Lucas numbers at consecutive and nonconsecutive terms and the products of these (p, q)-numbers with Mersenne numbers at consecutive and nonconsecutive terms.

The feasibility of considering quasi-Mersenne numbers represented in the form p = 3k + 2 is justi- fied. The first prime numbers represented in this form are 11, 29, 83, 6563, i.e. there exist specific Galois fields corresponding to such numbers. It is shown that computations in such Galois fields can be reduced to computations in finite algebraic rings, specifically, in rings of subtraction classes modulo 3k + 1. It is found that the numbers under consideration have a property similar to that possessed by the Mersenne prime numbers in binary representation. Specifically, when multiplying numbers of the considered type written in ternary representation by 3, there is a cyclic rearrangement of ternary symbols with a change of sign. The highlighted case is an exception: in rings of subtraction classes modulo 3k + 1 there exists a highlighted element whose multiplication by three modulo 3k + 1 remains unchanged. We discuss the pos- sibilities of application of the found property for realisation of computational systems operating in Galois fields and finite algebraic rings.

In order to strictly prove from the point of view of pure mathematics Goldbach's 1742 Goldbach conjecture and Hilbert's twinned prime conjecture in question 8 of his report to the International Congress of Mathematicians in 1900, and the French scholar Alfond de Polignac's 1849 Polignac conjecture, By using Euclid's principle of infinite primes, equivalent transformation principle, and the idea of normalization of set element operation, this paper proves that Goldbach's conjecture, twin primes conjecture and Polignac conjecture are completely correct. In order to strictly prove a conjecture about the solution of positive integers of indefinite equations proposed by French scholar Ferma around 1637 (usually called Ferma's last theorem) from the perspective of pure mathematics, this paper uses the general solution principle of functional equations and the idea of symmetric substitution, as well as the inverse method. It proves that Fermar's last theorem is completely correct.

This paper considers the AJPS-1 post-quantum cryptosystem.A feature of this cryptosystem is the use of arithmetic modulo Mersenne number, in particular, the AJPS cryptosystem uses relations for the Hamming weight of integers modulo Mersenne number.To create a modification of this cryptosystem by changing the metric, relations of the OSD metric for integers modulo Mersenne number were obtained.The paper describes the constructed modification of the AJPS-1 cryptosystem with a changed metric and analyses its advantages compared to the AJPS-1 cryptosystem.This modification allows to increase the variance of the decryption parameter, which improves the resistance of the cryptosystem to ciphertext-only (known ciphertext) attacks aimed at determining the private key.

In this paper, we present the determinantal identities of generalized Gaussian Fibonacci numbers. The generalized Gaussian Fibonacci sequence is defined by the recurrence relation. This was introduced by S. Pethe and A. F. Horadam. Also, we present its determinantal identities with classical numbers like gaussian Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, jacobsthal-Lucas, Bronze, Nickel and Mersenne numbers.

In this paper, we present the determinantal identities of generalized Gaussian Fibonacci numbers. The generalized Gaussian Fibonacci sequence is defined by the recurrence relation. This was introduced by S. Pethe and A. F. Horadam. Also, we present its determinantal identities with classical numbers like gaussian Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, jacobsthal-Lucas, Bronze, Nickel and Mersenne numbers.

In a previous paper, a new generalized definition of Mersenne numbers was proposed of the form (a n -(a -1) n ) called Global Generalized Mersenne numbers, or Generalized Mersenne numbers in short. For prime exponents n, Generalized Mersenne primes and composites are generated. In this paper, the properties and distributions of Generalized Mersenne composites are investigated. It is found that the distribution of composite Generalized Mersenne numbers follow simple laws demonstrated in three theorems, as composite GM a,n appear periodically in an infinite number of groups of pairs of solutions in a, embedded into each others. It is remarkable that the distribution of composite GM a,n is completely characterized once the first values of a yielding composite GM a,n are found, as composite GM a,n are spaced regularly, separated by intervals of values depending on their factors c 1 = 2nf 1 + 1. Three methods are presented to calculate composite GM a,n and applied for the first six prime exponents n from 3 to 17.

In this paper, we introduced Gaussian Fermat numbers and polynomials. We provided the Binet formula, generating functions, and the exponential generating function for these numbers and polynomials. Additionally, we derived several identities for these polynomials, including the Cassini identity, Catalan identity, Vajda identity, Halton identity, Gelin-Cesaro identity, and D’Ocagne’s identity. We demonstrated that Gaussian Fermat numbers and polynomials can also be obtained through matrix representations and discussed key propositions based on the fact that the determinants of these matrix representations are constant. Furthermore, we explored the relationship between Gaussian Fermat numbers, polynomials, Mersenne numbers, and Jacobsthal numbers. We also presented the Catalan, Binomial, and Binomial of Catalan transformations of the Gaussian Fermat sequence and polynomials. Finally, we introduced the generating function for the Catalan transformation of the Gaussian Fermat numbers and polynomials.