Given the $n$th prime number, is it possible to express explicitly a function $\phi:P \mapsto \mathbb{N}$, where $P$ is the set of all prime numbers, such that $\phi(p_n)$ gives the prime gap after $p_n$?
Does there exist an explicit formula that expresses $p_{n+1}-p_{n}$ in terms of $p_n$, the $n$th prime number?
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number-theory
prime-numbers
analytic-number-theory
prime-gaps
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1Even if such function exists it must be very "inconvenient" to use. Mathematicians still do not know if there are infinitely many prime twins: so looking at this function it's not so easy even to tell if it's value can be 2 for an arbitrarily large argument. Must be some very ugly function, I do not think it exists. – 2017-01-19
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0$p_2-p_1=1$ and $p_{n+1}-p_{n}=2$ for other **twin primes**. – 2017-01-19
1 Answers
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No, I don't think so. Because otherwise we would have a function which gives us all the prime numbers. It is easily seen, that we would have $$p_{n + 1} = p_n + \phi(p_n)$$ and inductively we would get every prime number.