Using a greedy algorithm, one can match a -heavy prime to each -heavy prime (counting multiplicity) in such a way that for a small (in most cases one can make , and often one also has ). If we then replace in the factorization of by for each -heavy prime , this increases (and ...
Using a greedy algorithm, one can match a -heavy prime to each -heavy prime (counting multiplicity) in such a way that for a small (in most cases one can make , and often one also has ). If we then replace in the factorization of by for each -heavy prime , this increases (and ...
Both approaches have their strengths and weaknesses, depending on the problem to be solved, with generative AI being well-suited for tasks involving NLP and for the creation of new content, and traditional algorithms more effective for tasks involving rule-based processing and predetermined outcomes....
Problems cryptography solves In this day and age, millions of people use devices and are connected to each other. While these different users may trust each other, data and communications transferred between them may often travel along networks that cannot be trusted (such as the public internet,...
One of the goals of quantum computing research is to study which problems can be solved by a quantum computer faster than a classical computer and how large the speedup can be. One well-known example is the Grover's algorithm, which yield a polynomial speedup over the classical counterparts....
I've recently started solving problems on AtCoder and noticed that the quality of problems is exceptional. Obviously Codeforces also offers great problems, AtCoder's seem to be on another level. Does anyone know why this might be? One possible reason could be that AtCoder hosts contests less...
Classical cryptography—such as the Rivest–Shamir–Adleman (RSA) algorithm that’s widely used to secure data transmission—relies on the intractability of problems such as integer factorization or discrete logarithms. Many of these problems can be solved more efficiently using quantum computers. Optimi...
The main reason for computing error bounds is not to get precise bounds but rather to verify that the formula does not contain numerical problems. A final example of an expression that can be rewritten to use benign cancellation is (1 + x)n, where . This expression arises in financial ...
P Problems: These are problems classical computers can solve efficiently in polynomial time. An example of such problems is: Givennnumbers and another numberk, is there a number innthat’s larger thank? This problem can be solved easily in linear time. ...
one of those once-impossible problems is prime factorization. The mathematician Peter Shor showed in 1994 that a sufficiently powerful future quantum computer would be able to find the prime factors of integers much more easily than classical computers. Shor's algorithm was the first algorithm ever...