# Tagged Questions

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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### Reasoning about $z^n = x^m + y^m$

Let $z,n,x,y,m$ be positive integers with $z \ge 5$ and $m \ge 3$ and $m$ odd. Does it follow that: $z$ cannot be prime if $p \ge 5$ and $p | z$, then either $p > m$ or $p|m$ Here is my ...
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### About $x(\ln(x\ln(x))-1)<p_x<x\ln(x\ln(x)), x>5$ and better results

I need some tips about this: It has been proved that (1) $$x(\ln(x\ln(x))-1)<p_x<x\ln(x\ln(x)),\quad x>5$$ Is there a better results? Thanks!
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### About $\pi(x)<li(x)-\frac{1}{3}\frac{\sqrt{x}}{\log(x)}\log(\log(\log(x)))$

Today I found that in 1914, Littlewood proved that (1) there are arbitrarily large values of $x$ for which $$\pi(x)<li(x)-\frac{1}{3}\frac{\sqrt{x}}{\log(x)}\log(\log(\log(x)))$$ First: Is ...
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### Has this partial result about Legendre's Conjecture been proved?

I'd like to know if it has been proved this "partial result" about the Legendre's Conjecture: (1) There are infinitely many $n$ such that there's a prime in $(n^2, (n+1)^2)$ Thanks!
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### Prove that if $n > 2$ then between $n$ and $n!$ is at least one prime. [duplicate]

Prove that if $n > 2$ then between $n$ and $n!$ is at least one prime. Ok I can see that it's obviously true, but what to use to prove it?
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### What is the value of $\sum_{p\le x} 1/p^2$?

My question is, what is the value of $$\sum_{p\le x} \frac{1}{p^2}?$$ More generally, what is the value of $$\sum_{p\le x} \frac{1}{p^n}?$$ How can we find it? For $\sum_{p\le x} 1/p$ the idea was ...
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### Curve profile for the logarithm-integral sum term of Riemann explicit formula?

I am considering the following term from the Riemann explicit formula (see here >>>): $$\sum_{\rho(\Im>0)}{\mathrm{li}(x^\rho)}$$ with $\rho$ non-trivial zeros of $\zeta$-function. I have a plot ...
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### How to calculate the $i$-th element in the sequence of prime numbers?

The sequence of prime numbers is the set of prime numbers in their natural order (that is, $2, 3, 5, 7, 11, 13, 17,...$). The German wikipedia entry on sequences states the following: Given $i$, ...
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### Explicit formulas for primitive roots?

For a Fermat prime or an "upper" Sophie Germain prime a primitive root is explicitly known. Are there further results when the factorization of p-1 is known? Is it unlikely that we ever get explicit ...