# Questions tagged [prime-gaps]

The difference of two prime consecutive prime numbers is the prime gap. $g_i := p_{i+1} - p_i$.

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### Finding a special sequence related to primes

Let $(g_{k})_{k≥1}$ be the sequence of primes gaps (https://en.wikipedia.org/wiki/Prime_gap#Lower_bounds). I am asking about the possibility of finding a real sequence $(x_{k})_{k≥1}$ with the ...
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### Can we deduce that there is infinitely many indices $n$ such that the period length of $1/(2^{2^n}+1)$ is strictly less then $2^{2^n}$.

In this page (http://mathworld.wolfram.com/FermatPrime.html) we have the following result: $2^{2^n}+1$ is a Fermat prime if and only if the period length of $1/(2^{2^n}+1)$ is equal to $2^{2^n}$. In ...
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### A weaker version of the Firoozbakht's conjecture

Firoozbakht's conjecture states that: $p_{k}^{1/k}$ is a strictly decreasing function of $k≥1$. Here $p_{k}$ is the sequence of primes. I know that is statement is not yet proved. But I am asking on ...
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### A weaker version of the Andrica's conjecture

Andrica's conjecture states that: For every pair of consecutive prime numbers $p_{k}$ and $p_{k+1}$, we have : $$\sqrt{p_{k+1}}-\sqrt{p_{k}}<1\quad\quad \color{#2d0}{\text{(1.)}}$$ I know that is ...
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### A trivial proof of Bertrand's postulate

Write the integers from any $n$ through $0$ descending in a column, where $n \geq 2$, and begin a second column with the value $2n$. For each entry after that, if the two numbers on that line share a ...