The algorithm I'm referring to is one of the most fundamental primality checks: For a number, $n$, check if it is divisible by some odd number, $k$, less than or equal to $\sqrt{n}$. Assume $n$ is a fixed size and that all basic arithmetic operations (add, subtract, multiply, divide, remainder) run in $O(1)$. I don't see why this algorithm will not be $O(\sqrt{n})$, but apparently it's not since PRIMES was considered an NP problem up until the AKS primality check.
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The problem with your statement is that you assume that all the basic arithmetic operations are $O(1)$ and that the amount of numbers you will have to check is $O(\sqrt{n})$, relative to the size of the input.
This is wrong because $\text{PRIMES}$ is the language of all the prime numbers, given in a binary format. So an input of length $n$ will be a number of size $O(2^n)$. Then, if the input is some number $x$, in order to check all the numbers up to $\sqrt x$ you will have to go over $O(\sqrt{2^n})$ numbers, which is $O(2^{n/2})$ numbers. Even with $O(1)$ operations this will not take a polynomial amount of time.
Mickey
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1Ah, so $n$ has to be the actual size of the input, not the input itself? – Badr B Jun 02 '18 at 08:51
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1@BadrB exactly :) – Mickey Jun 02 '18 at 12:03