Some Interesting Mathematics about Prime Factorization

1975 ◽  
Vol 68 (1) ◽  
pp. 73-74
Author(s):  
Joe Tom Rodgers
Keyword(s):  
Prime Factorization ◽  
Factoring Polynomials ◽  
Greatest Common Divisors ◽  
School Careers

Throughout their school careers, students are taught how to find the prime factorization of numbers. Most students readily learn the technique and find the process interesting. Most teachers then teach the ways in which prime factorizations can aid in factoring polynomials and finding greatest common divisors as well as least common multiples.

scholarly journals New Semi-Prime Factorization and Application in Large RSA Key Attacks

2021 ◽  
Vol 1 (4) ◽  
pp. 660-674
Author(s):  
Anthony Overmars ◽  
Sitalakshmi Venkatraman
Keyword(s):  
Solution Space ◽  
Public Key Cryptography ◽  
Factorization Method ◽  
Future Research ◽  
Opposite Parity ◽  
Prime Factorization ◽  
Factorization Problem ◽  
Pythagorean Triples ◽  
Greatest Common Divisors ◽  
Pythagorean Quadruples

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.

Experiment on Object Persistency in Distributed Parallel Processing for Prime Factorization

2004 ◽  
Vol 14 (3) ◽  
pp. 50-54
Author(s):  
Hidenobu TSURUSAWA ◽  
Ichiro IIMURA ◽  
Shigeru NAKAYAMA
Keyword(s):  
Parallel Processing ◽  
Prime Factorization

scholarly journals Continuous assessment and matriculation examination marks – An empirical examination

2015 ◽  
Vol 5 (2) ◽  
pp. 17 ◽  
Author(s):  
Servaas Van der Berg ◽  
Debra Shepherd
Keyword(s):  
South African ◽  
Career Planning ◽  
National Curriculum ◽  
Continuous Assessment ◽  
Empirical Examination ◽  
Sense Of Security ◽  
False Sense ◽  
Matriculation Examination ◽  
School Careers ◽  
Examination Preparation

<p>This study analyses information and feedback from matriculation level continuous assessment in the South African education system. Continuous assessment (CASS) at the time carried a 25% weight in the final matriculation (Grade 12) mark, and it provides feedback that affects examination preparation and effort. Weak assessment in schools sends wrong signals to students that may have important consequences for the way they approach the final examination. Moreover, similarly wrong signals earlier in their school careers may also have affected their subject choice and career planning.<br />This study compares CASS data to the externally assessed matric exam marks for a number of subjects. There are two signalling dimensions to inaccurate assessments: (i) Inflated CASS marks can give students a false sense of security and lead to diminished exam effort. (ii) A weak correlation between CASS and the exam marks could mean poor signalling in another dimension: Relatively good students may get relatively low CASS marks. Such low correlations indicate poor assessment reliability, as the examination and continuous assessment should both be testing mastery of the same national curriculum. The paper analyses the extent of each of these dimensions of weak signalling in South African schools and draws disturbing conclusions for a large part of the school system.</p>

Plan B: B-School [Careers]

IEEE Spectrum ◽  
2011 ◽  
Vol 48 (2) ◽  
pp. 22-23
Author(s):  
Prachi Patel
Keyword(s):  
Plan B ◽  
School Careers
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