How to Create an Efficient Infinite Prime Number Generator in Python?

How to implement an efficient infinite generator of prime numbers in Python?

The task is to create an efficient infinite generator of prime numbers in Python. The key term here is "INFINITE," implying that the generator can produce an endless stream of prime numbers without knowing how many will be consumed.

The Proposed Solution: Three Optimized Sieve Functions

The provided code offers three optimized Sieve functions: erat2, erat2a, and erat3. Each function gradually improves upon the previous one in terms of speed and efficiency.

erat2a

The erat2a function is a modification of the erat2 function, which is not included here. By optimizing the oddity check and reducing the number of steps taken for candidates, erat2a speeds up the generation by 20-25%.

erat3

erat3 further enhances erat2a by leveraging the fact that modulo 30, all primes except for 2, 3, and 5 result in only eight numbers (1, 7, 11, 13, 17, 19, 23, 29). This allows erat3 to filter out non-candidate odd numbers more efficiently, resulting in a 35-40% speed increase.

Benchmark Results

Benchmarks on different hardware configurations demonstrate the performance improvements:

On an Atom 330 Ubuntu 9.10 server, erat3 outperforms erat2 and erat2a on both Python 2 and 3.

On an AMD Geode LX Gentoo home server, erat3 again shows significant performance gains, outperforming the other functions on both Python 2 and 3.

These optimizations in the Sieve functions provide a significant advantage in generating prime numbers efficiently, making them suitable for various mathematical and computational applications.

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