You’re not quite implementing the correct algorithm: In your first example, primes_sieve doesn’t maintain a list of primality flags to strike/unset (as in the algorithm), but instead resizes a list of integers continuously, which is very expensive: removing an item from a list requires shifting all subsequent items down by one. In the second example, … Read more
Yes, this method can be used in cryptography. RSA encryption involves the finding of huge prime numbers, sometimes on the order of 1024 bits (about 300 digits). The security of RSA depends on the fact that factoring a number that consists of 2 of these prime numbers multiplied together is extremely difficult and time consuming. … Read more
Use the estimate pi(n) = n / log(n) for the number of primes up to n to find a limit, and then use a sieve. The estimate underestimates the number of primes up to n somewhat, so the sieve will be slightly larger than necessary, which is ok. This is my standard Java sieve, computes … Read more
This question was the first link that popped up when I googled “python prime factorization”. As pointed out by @quangpn88, this algorithm is wrong (!) for perfect squares such as n = 4, 9, 16, … However, @quangpn88’s fix does not work either, since it will yield incorrect results if the largest prime factor occurs … Read more
You said you understand this part, but just to emphasize, the String generated has a length equal to the number supplied. So the string has three characters if and only if n == 3. .? The first part of the regex says, “any character, zero or one times”. So basically, is there zero or one … Read more
The fastest algorithm for general prime testing is AKS. The Wikipedia article describes it at lengths and links to the original paper. If you want to find big numbers, look into primes that have special forms like Mersenne primes. The algorithm I usually implement (easy to understand and code) is as follows (in Python): def … Read more
Prime numbers are chosen to best distribute data among hash buckets. If the distribution of inputs is random and evenly spread, then the choice of the hash code/modulus does not matter. It only has an impact when there is a certain pattern to the inputs. This is often the case when dealing with memory locations. … Read more
Most basic and general explanation: cryptography is all about number theory, and all integer numbers (except 0 and 1) are made up of primes, so you deal with primes a lot in number theory. More specifically, some important cryptographic algorithms such as RSA critically depend on the fact that prime factorization of large numbers takes … Read more
A very fast implementation of the Sieve of Atkin is Dan Bernstein’s primegen. This sieve is more efficient than the Sieve of Eratosthenes. His page has some benchmark information.
Warning: timeit results may vary due to differences in hardware or version of Python. Below is a script which compares a number of implementations: ambi_sieve_plain, rwh_primes, rwh_primes1, rwh_primes2, sieveOfAtkin, sieveOfEratosthenes, sundaram3, sieve_wheel_30, ambi_sieve (requires numpy) primesfrom3to (requires numpy) primesfrom2to (requires numpy) Many thanks to stephan for bringing sieve_wheel_30 to my attention. Credit goes to Robert … Read more