Relevant bibliographies by topics / Fermat's factorization

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  2. Dissertations / Theses
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  4. Conference papers

Academic literature on the topic 'Fermat's factorization'

Author: Grafiati
Published: 4 June 2025
Last updated: 15 July 2025

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Journal articles on the topic "Fermat's factorization"

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Aminudin, Aminudin, and Eko Budi Cahyono. "A Practical Analysis of the Fermat Factorization and Pollard Rho Method for Factoring Integers." Lontar Komputer : Jurnal Ilmiah Teknologi Informasi 12, no. 1 (2021): 33. http://dx.doi.org/10.24843/lkjiti.2021.v12.i01.p04.

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The development of public-key cryptography generation using the factoring method is very important in practical cryptography applications. In cryptographic applications, the urgency of factoring is very risky because factoring can crack public and private keys, even though the strength in cryptographic algorithms is determined mainly by the key strength generated by the algorithm. However, solving the composite number to find the prime factors is still very rarely done. Therefore, this study will compare the Fermat factorization algorithm and Pollard rho by finding the key generator public key algorithm's prime factor value. Based on the series of test and analysis factoring integer algorithm using Fermat's Factorization and Pollards' Rho methods, it could be concluded that both methods could be used to factorize the public key which specifically aimed to identify the prime factors. During the public key factorizing process within 16 bytes – 64 bytes, Pollards' Rho's average duration was significantly faster than Fermat's Factorization.
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James, Joseph. "UNIQUE FACTORIZATION FERMAT'S LAST THEOREM BEAL'S CONJECTURE." Journal of Progressive Research in Mathematics 10, no. 1 (2016): 1434–39. https://doi.org/10.5281/zenodo.3976651.

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In this paper the following statememt of Fermat\rq{}s Last Theorem is proved.  If  $x, y, z$ are positive integers$\pi$ is an odd prime and  $z^\pi=x^\pi+y^\pi, x, y, z$ are all even. Also, in this paper, is proved (Beal\rq{}s conjecture): The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z, $ with $\xi, \mu, \nu$ primes at least $3.
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Kritsanapong, Somsuk. "The new integer factorization algorithm based on fermat's factorization algorithm and euler's theorem." International Journal of Electrical and Computer Engineering (IJECE) 10, no. 2 (2020): 1469–76. https://doi.org/10.11591/ijece.v10i2.pp1469-1476.

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Although, Integer Factorization is one of the hard problems to break RSA, many factoring techniques are still developed. Fermat’s Factorization Algorithm (FFA) which has very high performance when prime factors are close to each other is a type of integer factorization algorithms. In fact, there are two ways to implement FFA. The first is called FFA-1, it is a process to find the integer from square root computing. Because this operation takes high computation cost, it consumes high computation time to find the result. The other method is called FFA-2 which is the different technique to find prime factors. Although the computation loops are quite large, there is no square root computing included into the computation. In this paper, the new efficient factorization algorithm is introduced. Euler’s theorem is chosen to apply with FFA to find the addition result between two prime factors. The advantage of the proposed method is that almost of square root operations are left out from the computation while loops are not increased, they are equal to the first method. Therefore, if the proposed method is compared with the FFA-1, it implies that the computation time is decreased, because there is no the square root operation and the loops are same. On the other hand, the loops of the proposed method are less than the second method. Therefore, time is also reduced. Furthermore, the proposed method can be also selected to apply with many methods which are modified from FFA to decrease more cost.
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Oduor, Maurice, Olege Fanuel, Aywa Shem, and Okaka A. Colleta. "CHARACTERIZATION OF CODES OF IDEALS OF THE POLYNOMIAL RING F30 2 [x] mod ô€€€ x30 ô€€€ 1 FOR ERROR CONTROL IN COMPUTER APPLICATONS." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 5 (2016): 6238–47. http://dx.doi.org/10.24297/jam.v12i5.260.

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The study of ideals in algebraic number system has contributed immensely in preserving the notion of unique factorization in rings of algebraic integers and in proving Fermat's last Theorem. Recent research has revealed that ideals in Noethe-rian rings are closed in polynomial addition and multiplication.This property has been used to characterize the polynomial ring Fn 2 [x] mod (xn 1) for error control. In this research we generate ideals of the polynomial ring using GAP software and characterize the polycodewords using Shannon's Code region and Manin's bound.
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Malik, M. Aslam, and M. Khalid Mahmood. "On Simple Graphs Arising from Exponential Congruences." Journal of Applied Mathematics 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/292895.

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We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integersaandb, letG(n)denote the graph for whichV={0,1,…,n−1}is the set of vertices and there is an edge betweenaandbif the congruenceax≡b (mod n)is solvable. Letn=p1k1p2k2⋯prkrbe the prime power factorization of an integern, wherep1<p2<⋯<prare distinct primes. The number of nontrivial self-loops of the graphG(n)has been determined and shown to be equal to∏i=1r(ϕ(piki)+1). It is shown that the graphG(n)has2rcomponents. Further, it is proved that the componentΓpof the simple graphG(p2)is a tree with root at zero, and ifnis a Fermat's prime, then the componentΓϕ(n)of the simple graphG(n)is complete.
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Babindamana, Regis Freguin, Gilda Rech Bansimba, and Basile Guy Richard Bossoto. "Lattice Points on the Fermat Factorization Method." Journal of Mathematics 2022 (January 28, 2022): 1–18. http://dx.doi.org/10.1155/2022/6360264.

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In this paper, we study algebraic properties of lattice points of the arc on the conics x 2 − d y 2 = N especially for d = 1 , which is the Fermat factorization equation that is the main idea of many important factorization methods like the quadratic field sieve, using arithmetical results of a particular hyperbola parametrization. As a result, we present a generalization of the forms, the cardinal, and the distribution of its lattice points over the integers. In particular, we prove that if N − 6 ≡ 0 mod 4 , Fermat’s method fails. Otherwise, in terms of cardinality, it has, respectively, 4, 8, 2 α + 1 , 1 − δ 2 p i 2 n + 1 , and 2 ∏ i = 1 n α i + 1 lattice pointts if N is an odd prime, N = N a × N b with N a and N b being odd primes, N = N a α with N a being prime, N = ∏ i = 1 n p i with p i being distinct primes, and N = ∏ i = 1 n N i α i with N i being odd primes. These results are important since they provide further arithmetical understanding and information on the integer solutions revealing factors of N . These results could be particularly investigated for the purpose of improving the underlying integer factorization methods.
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Overmars, Anthony, and Sitalakshmi Venkatraman. "New Semi-Prime Factorization and Application in Large RSA Key Attacks." Journal of Cybersecurity and Privacy 1, no. 4 (2021): 660–74. http://dx.doi.org/10.3390/jcp1040033.

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Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.
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Candra, Ade, Mohammad Andri Budiman, and Dian Rachmawati. "On Factoring The RSA Modulus Using Tabu Search." Data Science: Journal of Computing and Applied Informatics 1, no. 1 (2017): 30–37. http://dx.doi.org/10.32734/jocai.v1.i1-65.

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It is intuitively clear that the security of RSA cryptosystem depends on the hardness of factoring a very large integer into its two prime factors. Numerous studies about integer factorization in the field of number theory have been carried out, and as a result, lots of exact factorization algorithms, such as Fermat’s factorization algorithm, quadratic sieve method, and Pollard’s rho algorithm have been found. The factorization problem is in the class of NP (non-deterministic polynomial time). Tabu search is a metaheuristic in the field of artificial intelligence which is often used to solve NP and NP-hard problems; the result of this method is expected to be close-to-optimal (suboptimal). This study aims to factorize the RSA modulus into its two prime factors using tabu search by conducting experiments in Python programming language and to compare its time performance with an exact factorization algorithm, i.e. Pollard’s algorithm. The primality test is done with Lehmann’s algorithm.
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Raghunandan, K. R., Aithal Ganesh, Shetty Surendra, and K. Bhavya. "Key Generation Using Generalized Pell’s Equation in Public Key Cryptography Based on the Prime Fake Modulus Principle to Image Encryption and Its Security Analysis." Cybernetics and Information Technologies 20, no. 3 (2020): 86–101. http://dx.doi.org/10.2478/cait-2020-0030.

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AbstractRSA is one among the most popular public key cryptographic algorithm for security systems. It is explored in the results that RSA is prone to factorization problem, since it is sharing common modulus and public key exponent. In this paper the concept of fake modulus and generalized Pell’s equation is used for enhancing the security of RSA. Using generalized Pell’s equation it is explored that public key exponent depends on several parameters, hence obtaining private key parameter itself is a big challenge. Fake modulus concept eliminates the distribution of common modulus, by replacing it with a prime integer, which will reduce the problem of factorization. It also emphasizes the algebraic cryptanalysis methods by exploring Fermat’s factorization, Wiener’s attack, and Trial and division attacks.
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Omollo, Richard, and Arnold Okoth. "Large Semi Primes Factorization with Its Implications to RSA Cryptosystems." BOHR International Journal of Smart Computing and Information Technology 3, no. 1 (2020): 1–8. http://dx.doi.org/10.54646/bijscit.011.

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RSA’s strong cryptosystem works on the principle that there are no trivial solutions to integer factorization. Furthermore, factorization of very large semi primes cannot be done in polynomial time when it comes to the processing power of classical computers. In this paper, we present the analysis of Fermat’s Last Theorem and Arnold’s Theorem. Also highlighted include new techniques such as Arnold’s Digitized Summation Technique (A.D.S.T.) and a top-to-bottom, bottom-to-top approach search for the prime factors. These drastically reduce the time taken to factorize large semi primes as for the case in RSA Cryptosystem.
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Dissertations / Theses on the topic "Fermat's factorization"

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Hammad, Yousef Bani. "Novel methods for primality testing and factoring." Thesis, Queensland University of Technology, 2005. https://eprints.qut.edu.au/16142/1/Yousef_Bani_Hammad_Thesis.pdf.

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From the time of the Greeks, primality testing and factoring have fascinated mathematicians, and for centuries following the Greeks primality testing and factorization were pursued by enthusiasts and professional mathematicians for their intrisic value. There was little practical application. One example application was to determine whether or not the Fermat numbers, that is, numbers of the form F;, = 2'" + 1 were prime. Fermat conjectured that for all n they were prime. For n = 1,2,3,4, the Fermat numbers are prime, but Euler showed that F; was not prime and to date no F,, n 2 5 has been found to be prime. Thus, for nearly 2000 years primality testing and factorization was largely pure mathematics. This all changed in the mid 1970's with the advent of public key cryptography. Large prime numbers are used in generating keys in many public key cryptosystems and the security of many of these cryptosystems depends on the difficulty of factoring numbers with large prime factors. Thus, the race was on to develop new algorithms to determine the primality or otherwise of a given large integer and to determine the factors of given large integers. The development of such algorithms continues today. This thesis develops both of these themes. The first part of this thesis deals with primality testing and after a brief introduction to primality testing a new probabilistic primality algorithm, ALI, is introduced. It is analysed in detail and compared to Fermat and Miller-Rabin primality tests. It is shown that the ALI algorithm is more efficient than the Miller-Rabin algorithm in some aspects. The second part of the thesis deals with factoring and after looking closely at various types of algorithms a new algorithm, RAK, is presented. It is analysed in detail and compared with Fermat factorization. The RAK algorithm is shown to be significantly more efficient than the Fermat factoring algorithm. A number of enhancements is made to the basic RAK algorithm in order to improve its performance. The RAK algorithm with its enhancements is known as IMPROVEDRAK. In conjunction with this work on factorization an improvement to Shor's factoring algorithm is presented. For many integers Shor's algorithm uses a quantum computer multiple times to factor a composite number into its prime factors. It is shown that Shor's alorithm can be modified in a way such that the use of a quantum computer is required just once. The common thread throughout this thesis is the application of factoring and primality testing techniques to integer types which commonly occur in public key cryptosystems. Thus, this thesis contributes not only in the area of pure mathematics but also in the very contemporary area of cryptology.
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Hammad, Yousef Bani. "Novel Methods for Primality Testing and Factoring." Queensland University of Technology, 2005. http://eprints.qut.edu.au/16142/.

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From the time of the Greeks, primality testing and factoring have fascinated mathematicians, and for centuries following the Greeks primality testing and factorization were pursued by enthusiasts and professional mathematicians for their intrisic value. There was little practical application. One example application was to determine whether or not the Fermat numbers, that is, numbers of the form F;, = 2'" + 1 were prime. Fermat conjectured that for all n they were prime. For n = 1,2,3,4, the Fermat numbers are prime, but Euler showed that F; was not prime and to date no F,, n 2 5 has been found to be prime. Thus, for nearly 2000 years primality testing and factorization was largely pure mathematics. This all changed in the mid 1970's with the advent of public key cryptography. Large prime numbers are used in generating keys in many public key cryptosystems and the security of many of these cryptosystems depends on the difficulty of factoring numbers with large prime factors. Thus, the race was on to develop new algorithms to determine the primality or otherwise of a given large integer and to determine the factors of given large integers. The development of such algorithms continues today. This thesis develops both of these themes. The first part of this thesis deals with primality testing and after a brief introduction to primality testing a new probabilistic primality algorithm, ALI, is introduced. It is analysed in detail and compared to Fermat and Miller-Rabin primality tests. It is shown that the ALI algorithm is more efficient than the Miller-Rabin algorithm in some aspects. The second part of the thesis deals with factoring and after looking closely at various types of algorithms a new algorithm, RAK, is presented. It is analysed in detail and compared with Fermat factorization. The RAK algorithm is shown to be significantly more efficient than the Fermat factoring algorithm. A number of enhancements is made to the basic RAK algorithm in order to improve its performance. The RAK algorithm with its enhancements is known as IMPROVEDRAK. In conjunction with this work on factorization an improvement to Shor's factoring algorithm is presented. For many integers Shor's algorithm uses a quantum computer multiple times to factor a composite number into its prime factors. It is shown that Shor's alorithm can be modified in a way such that the use of a quantum computer is required just once. The common thread throughout this thesis is the application of factoring and primality testing techniques to integer types which commonly occur in public key cryptosystems. Thus, this thesis contributes not only in the area of pure mathematics but also in the very contemporary area of cryptology.
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Podhorský, Jiří. "Integer Factorization on the GPU." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2014. http://www.nusl.cz/ntk/nusl-412908.

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This work deals with factorization, a decomposition of composite numbers on prime numbers and possibilities of its parallelization. It summarizes also the best known algorithms for factoring and most popular platforms for the implementation of these algorithms on the graphics card. The main part of the thesis deals with the design and implementation of hardware acceleration current fastest algorithm on the graphics card by using the OpenCL framework. Subsequently, the work provides a comparison of speeds accelerated algorithm implemented in this work with other versions of the best known algorithms for factoring, processed serially. In conclusion, the work discussed length of RSA key needed for safe operation without the possibility of breaking in real time interval.
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Breitenbacher, Dominik. "Paralelizace faktorizace celých čísel z pohledu lámání RSA." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2015. http://www.nusl.cz/ntk/nusl-234905.

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This paper follows up the factorization of integers. Factorization is the most popular and used method for RSA cryptoanalysis. The SIQS was chosen as a factorization method that will be used in this paper. Although SIQS is the fastest method (up to 100 digits), it can't be effectively computed at polynomial time, so it's needed to look up for options, how to speed up the method as much as possible. One of the possible ways is paralelization. In this case OpenMP was used. Other possible way is optimalization. The goal of this paper is also to show, how easily is possible to use paralelizion and thanks to detailed analyzation the source codes one can reach relatively large speed up. Used method of iterative optimalization showed itself as a very effective tool. Using this method the implementation of SIQS achieved almost 100 multiplied speed up and at some parts of the code even more.
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Brinkmann, Daniel. "Hilbert-Kunz functions of surface rings of type ADE." Doctoral thesis, 2013. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2013082711496.

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We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.
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Book chapters on the topic "Fermat's factorization"

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Bressoud, David M. "Fermat, Euler, and Pseudoprimes." In Factorization and Primality Testing. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-4544-5_3.

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Bressoud, David M. "Factorization Techniques from Fermat to Today." In Factorization and Primality Testing. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-4544-5_5.

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Wu, Mu-En, Raylin Tso, and Hung-Min Sun. "On the Improvement of Fermat Factorization." In Network and System Security. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-34601-9_29.

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Ali, Fahed, and Mehwish Mustafa. "IMMFV2: Improved Modified Fermat Factorization Algorithm." In Communications in Computer and Information Science. Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-19968-4_12.

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Somsuk, Kritsanapong, and Sumonta Kasemvilas. "Possible Prime Modified Fermat Factorization: New Improved Integer Factorization to Decrease Computation Time for Breaking RSA." In Advances in Intelligent Systems and Computing. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-06538-0_32.

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Balasubramanian, Kannan, and Ahmed Mahmoud Abbas. "Integer Factoring Algorithms." In Algorithmic Strategies for Solving Complex Problems in Cryptography. IGI Global, 2018. http://dx.doi.org/10.4018/978-1-5225-2915-6.ch017.

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Most cryptographic systems are based on an underlying difficult problem. The RSA cryptosystem and many other cryptosystems rely on the fact that factoring a large composite number into two prime numbers is a hard problem. The are many algorithms for factoring integers. This chapter presents some of the basic algorithms for integer factorization like the Trial Division, Fermat's Algorithm. Pollard's Rho Method, Pollard's p-1 method and the Elliptic Curve Method. The Number Field Sieve algorithm along with Special Number field Sieve and the General Number Field Sieve are also used in factoring large numbers. Other factoring algorithms discussed in this chapter are the Continued Fractions Algorithms and the Quadratic Sieve Algorithm.
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Reilly, Norman R. "Elliptic Curves." In Introduction to Applied Algebraic Systems. Oxford University PressNew York, NY, 2009. http://dx.doi.org/10.1093/oso/9780195367874.003.0008.

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Abstract Although elliptic curves have been studied for many decades and a wealth of information was gained about them, it is only relatively recently that they have gained a certain level of “notoriety”. This has been the result of the application of the theory of elliptic curves to such things as Fermat’s Last Theorem, cryptography, and integer factorization. In this chapter we develop somebasic properties of elliptic curves, particularly thegroup structureon an elliptic curve. There are other situations, and also simpler situations, when it is possibleto definean algebraic structureon a set of solutions to an equation. One simpler example where one is dealing exclusively with integer solutions occurs in the study of Pell’s Equation. This is treated in section 7.7, but you are welcome to digress to that section at this stage, if you wish to do so. To understand the critical intersection properties of elliptic curves, it is necessary to consider them in projective space. For this reason, we devote some time to introducing and working with projective spaces and, in particular, the projective plane.
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Conference papers on the topic "Fermat's factorization"

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Karpinski, Mikolaj, Stepan Ivasiev, Ihor Yakymenko, Mykhajlo Kasianchuk, and Tomasz Gancarczyk. "Advanced method of factorization of multi-bit numbers based on Fermat's theorem in the system of residual classes." In 2016 16th International Conference on Control, Automation and Systems (ICCAS). IEEE, 2016. http://dx.doi.org/10.1109/iccas.2016.7832500.

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