Let a be an ideal of Noetherian ring R and M a finitely generated R-module such that cd(a,M)=c. In this paper, we investigate AttR(Hac(M)). Among other things, it is shown that ...Max{p∈SuppRM|cd(a,R/p)=c}⊆AttR(Hac(M)). We also show that AttR(Hac(M))={p∈SuppRM|cd(a,R/p)=c, p=AnnR(Hac(R/p))} and {p∈SuppRM|cd(a,R/p)=c,dimR/p−1≤cd(a,R/p)≤dimR/p}⊆AttR(Hac(M)). Finally, we prove that if (R,m) is a local ring and dimR/a=1 then AttR(Hac(M))={p∈SuppRM|cd(a,R/p)=cd(a,M)}. Then by using this, it is shown that if (R,m) is a local ring then {p∈SuppRM|cd(a,R/p)=c,dimR/(a+p)=1}⊆AttR(Hac(M)).
We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this ...reduced curve is cyclic.
The RSA (Rivest–Shamir–Adleman) cryptosystem is an asymmetric public key cryptosystem popular for its use in encryptions and digital signatures. However, the Wiener’s attack on the RSA cryptosystem ...utilizes continued fractions, which has generated much interest in developing competitive factoring algorithms. A general-purpose integer factorization method first proposed by Lehmer and Powers formed the basis of the well-known Continued Fraction Factorization (CFRAC) method. Recent work on the one line factoring algorithm by Hart and its connection with Lehman’s factoring method have motivated this paper. The emphasis of this paper is to explore the representations of PQ as continued fractions and the suitability of lower ordered convergences as representations of ab. These simpler convergences are then prescribed to Hart’s one line factoring algorithm. As an illustration, we demonstrate the working of our approach with two numbers: one smaller number and another larger number occupying 95 bits. Using our method, the fourth convergence finds the factors as the solution for the smaller number, while the eleventh convergence finds the factors for the larger number. The security of the RSA public key cryptosystem relies on the computational difficulty of factoring large integers. Among the challenges in breaking RSA semi-primes, RSA250, which is an 829-bit semi-prime, continues to hold a research record. In this paper, we apply our method to factorize RSA250 and present the practical implementation of our algorithm. Our approach’s theoretical and experimental findings demonstrate the reduction of the search space and a faster solution to the semi-prime factorization problem, resulting in key contributions and practical implications. We identify further research to extend our approach by exploring limitations and additional considerations such as the difference of squares method, paving the way for further research in this direction.
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.
Let f(x) be a polynomial with integer coefficients. We say that the prime p is a prime divisor of f(x) if p divides f(m) some integer m. For each positive integer n, we give an explicit construction of ...a polynomial all of whose prime divisors are ±1 modulo (8n + 4). Consequently, this specific polynomial serves as an "Euclidean" polynomial for the Euclidean proof of Dirichlet's theorem on primes in the arithmetic progression ±1 (mod 8n + 4). Let
be a finite field with p
2
elements. We use that the multiplicative group of
is cyclic in our proof.
For any k≥1, this paper studies the number of polynomials having k irreducible factors (counted with or without multiplicities) in Fqt among different arithmetic progressions. We obtain asymptotic ...formulas for the difference of counting functions uniformly for k in a certain range. In the generic case, the bias dissipates as the degree of the modulus or k gets large, but there are cases when the bias is extreme. In contrast to the case of products of k prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon.
Number theorists believe that primes play a central role in Number theory and that solving problems related to primes could lead to the resolution of many other unsolved conjectures, including the ...prime k-tuples conjecture. This paper aims to demonstrate the existence of this conjecture for admissible k-tuples in a positive proportion. The authors achieved this by refining the methods of “Goldston, Pintz and Yildirim” and “James Maynard” for studying bounded gaps between primes and prime k-tuples. These refinements enabled to overcome the previous limitations and restrictions and to show that for a positive proportion of admissible k-tuples, there is the existence of the prime k-tuples conjecture holding for each “k”. The significance of this result is that it is unconditional which means it is proved without assuming any form of strong conjecture like the Elliott–Halberstam conjecture
Bicultural individuals often must select between different responses to a task implied by their dual identities. When one identity has been situationally primed, one strategy is to assimilate to the ...situational demand. However, another strategy is contrastive, to express their other identity. This second strategy has been called ethnic affirmation (Yang & Bond, 1980) and cultural contrast (Mok & Morris, 2013). It has been primarily explained as a defense against perceived threat to the specific identity that is excluded. The present studies posit that contrastive responses to cultural priming also may reflect a defense of biculturals' global self-integrity. We provide the first direct test of this mechanism by examining the effect of self-affirmation on the contrastive response tendency. In two experiments that varied self-affirmation and cultural priming, non-affirmed Chinese Americans showed contrastive responses—they exhibited greater extraversion in response to Chinese primes than to American primes in Study 1 or than to neutral primes in Study 2, but this pattern was significantly attenuated among Chinese Americans who engaged in self-affirmation. Theoretical implications are discussed.
In this paper problems 25, 86, 88, 105, 111, 137–142, and 184–185 from 12 are formalized, using the Mizar formalism 3, 1, 4. This is a continuation of the work from 5, 6, and 2 as suggested in 8. The ...automatization of selected lemmas from 11 proven in this paper as proposed in 9 could be an interesting future work.
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
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