Algorithmic complexity of primes

[DavidK]

Every number can be considered a bit string. For a bit string one can define some algorithmic complexity (the shortest algorithm/program that reproduces the desired bit string). Can something be said, in general, about difference in complexity for primes as compared to composites?

 

[Hurkyl]

There are roughly (2^n / (n log 2)) primes of bit length n or less. (Since $\pi(x) \approx x / \log x$)

The best possible compression scheme is to represent each prime by its index, so that we use the positive integers less than roughly (2^n / (n log 2)) to represent the primes. It takes roughly (n - lg n - lg log 2) bits to denote such numbers.

(A good descriptive complexity scheme will be worse than this by some fixed additive constant)

However, $n - \mathop{\mathrm{lg}} n - \mathop{\mathrm{lg}} \log 2 \in \Theta(n)$, so asymptotically, we haven't saved anything.

 

[tacman]

The algorithmic complexity of primes is a topic that has been studied extensively in computer science and mathematics. As stated in the content, every number can be represented as a bit string and the algorithmic complexity of a number refers to the shortest algorithm or program that can generate that particular bit string.

In general, it can be said that primes have a lower algorithmic complexity compared to composite numbers. This is because primes have a very specific and simple pattern, they can only be divided by 1 and themselves. This makes it easier to come up with a short algorithm or program to generate primes.

On the other hand, composite numbers have more complex patterns and can be divided by multiple numbers. This makes it harder to come up with a short algorithm or program to generate all composite numbers. In fact, the fastest known algorithm for finding all prime numbers up to a given number is significantly faster than the fastest known algorithm for finding all composite numbers up to the same number.

However, it is important to note that the complexity of primes and composites is not a definitive measure of their importance or significance. Primes have a unique and important role in mathematics, and the complexity of their generation does not diminish their value. Additionally, the complexity of a number does not necessarily reflect its difficulty in solving related problems, as some composite numbers can have simpler solutions compared to primes in certain mathematical problems.

In conclusion, while primes may have a lower algorithmic complexity compared to composites, this does not diminish the importance or significance of either type of number in mathematics. Both primes and composites have unique properties and play crucial roles in various mathematical concepts and problems.