# 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.

46 views

63 views

24 views

### Division/Remain by a Mersenne Prime

Is it possible to compute the integer division and remainder of an integer $x$ by a Mersenne prime $p$ using only bitwise operations?
713 views

28 views

### Generalization of Mill's theorem

Mill's theorem states that there exists a positive real number A such that the floor of the double exponential function $A^{3^n}$ are primes for all positive integers n. The value of A is ...
56 views

### A conjecture relating to Goldbach

I have a conjecture related to the strong Goldbach conjecture and the Goldbach function. It is that: for any $g(E)$, there are a finite number of even numbers which can be expressed as a sum of two ...
115 views

### Primes in the binomial transform of $[1, 1, 2, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, …]$.

This question is related to this sequence A139482. A commentator gives the following formula for $a_m$ $$a_m = {3m^2-9m+10 \above 1.5pt 2}$$ I have that you should consider the sequence $b_n =3n+2$ ...
52 views

### Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$.

Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$. I'm having a difficult time proving this problem. I was able to verify that it works for prime $n$ up to ...
3k views

### Can a complex number be prime?

I've been pondering over this question since a very long time. If a complex number can be prime then which parts of the complex number needs to be prime for the whole complex number to be prime.
33 views

### Can do you repeat these calculations combining the explicit formula and Nicolas criterion, on assumption of the Riemann Hypothesis?

I did easy calculations to get for $x=N_k=\prod_{n=1}^k p_k$ the kth primorial, combining the so-called explicit formula$\dagger$ for the second Chebyshev function and Nicolas criterion for the ...
66 views

### $p\in\mathbb P\iff\Big(2\leq k<\sqrt p\implies\gcd(k^2,p-k^2)=1\Big ),\;p>3$

This is sharper variant of A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$ It seems enough to test that for some sums: $p=m+n\implies\gcd(m,n)=1$, namely ...
23 views

### Common generator of units in finite prime fields

It is well known that the unit group of a finite field is cyclic. What can we say about the generators? Specifically I am interested in the following question: Suppose we fix a positive integer $a$, ...
38 views

### Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...
189 views

62 views

### It is possible to use the Zeta Function as primality test? [closed]

It is possible to use the Zeta Function as primality test? $$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s} = 1+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+ ...$$ Where can I find the non-trivial zeros ...
### Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $q\equiv m!\pmod p$ for some $2<m<p$?
Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...