In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these integers are further restricted to prime numbers, the process is called prime factorization.
When the numbers are very large, no efficient, non-quantum integer factorizationalgorithm is known; an effort by several researchers concluded in 2009, factoring a 232-digit number (RSA-768), utilizing hundreds of machines took two years and the researchers estimated that a 1024-bit RSA modulus would take about a thousand times as long. However, it has not been proven that no efficient algorithm exists. The presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.
Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, e.g., to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically.
Behind the fascinating technology and efficient operating model are Vikas Brahmavar and Gowthaman Desingh, former IT engineers who took grand leaps of faith to address a problem they witnessed every day ...Resistance to change is a major factor.
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Symptoms like severe headaches, vision problems, dizziness, or numbness require immediate medical attention. Risk factors include obesity, high blood pressure, ...
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They cited factors like the suspension of road dust due to vehicular traffic, increased construction activities, and waste burning as special concerns that have increased pollutants.
The issue is even affecting young people, who in a regionwide survey listed housing affordability and availability as a key factor impacting their decision to leave Hampton Roads. City governments are tackling the problem in different ways.
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While Dev and her colleagues did not speculate exactly how these factors could create problems, they did not rule out damage to brain tissue and the central nervous system.
Daily Press
30 Dec 2024
Daily Press
30 Dec 2024