# Questions tagged [prime-numbers]

Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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### Primality test for numbers of the form $\frac{a^p-1}{a-1}$?

Here is what I observed : Inspired by Lucas-Lehmer primality test, I think a made a primality test for numbers of the form $\frac{a^p-1}{a-1}$ but the test isn't perfect and there are some conditions ...
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### Closest multiples of 2 prime numbers [closed]

How can we find the distance between the closest multiples of 2 prime numbers (let's say a and b) smaller than a x b?
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### Primality test for $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ and $\frac{(10 \cdot 2^n)^m+1}{10 \cdot 2^n+1} - 2$

Here is what I observed: For $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ : Let $N$ = $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ : when $m$ is a number $m \ge 3$ and $n \ge 0$. Let the ...
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### What can be said about the primality of Zsigmondy numbers?

I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months. Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be the $n$-...
1 vote
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### A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
747 views

### A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis

Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
1 vote
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### Prime numbers formed by consecutive numbers

Let $x$ be an even number. Suppose that we concatenate it with its successor to form $x (x+1)$ (not multiplication, but concatenation). For example, if we start with $x = 2$, we would get $23$. If we ...
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### Sum of reciprocals of Sophie Germain primes

A prime $p$ is called a Sophie Germain prime if $2p+1$ is also prime: OEIS A005384. Whether there are an infinite number of such primes is unsolved. My question is: If there are an infinite number of ...
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### Estimates about prime numbers: a lemma in Bourgain's article

For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds: \begin{equation} |\{q\in \mathbb{N},q<Q: \text{all ...
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### Proofs in number theory that involve non-standard models of arithmetic

While reading an introductory text on model theory, I found it interesting that one can reformulate the famous conjectures about twin primes and Mersenne primes in terms of non-standard models of ...
531 views

### Solving functional equation $f(xy)=f(x+y)$ and Diophantine equations

It is well known that the only solution is $f$ a constant function. However, by putting some restrictions on the functional equation, we might get other solutions, with potential implications to ...
1 vote
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### No perfect patterns in the primes

The primes are equidistributed in the residue classes $1(\!\!\!\mod{4})$ and $3(\!\!\!\mod 4)$. We also know (for example, by Rubinstein-Sarnak) that the patterns cannot be eventually alternating, i.e....
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### Why is there an unexpected increase in the density of certain types of Goldbach primes?

Note: Posted in MO since it was unanswered in MSE. I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the ...
104 views

### Error term related to partial twin prime constant

The twin prime constant is defined as $$\Pi_2 = \prod_{\substack{p\in\mathcal{P}\\p>2}}\frac{p-2}{p-1}\left(1-\frac{1}{p}\right)^{-1},$$ where $\mathcal{P}$ is the set of primes. I'm interested ...
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### What is the importance of Polignac’s conjecture?

The twin-prime conjecture (also known as Polignac’s conjecture, 1846) states that there are infinitely many twin primes (pairs of primes that differ by 2; for example, 3 and 5, 5 and 7, 11 and 13, and ...
283 views

### How to explain a particular property of the second-to-last bits of primes?

Assuming that $i \geq 0$, let $p_i$ denote an $i$-th prime: $p_0 = 2, p_1 = 3, \ldots$ Then $b_i$ denotes the second-to-last bit of $p_i$, i.e. $b_i = \left\lfloor p_i/2 \right\rfloor \bmod 2.$ The ...
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### Prime numbers from another permutation

Related question: Prime numbers from permutation Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n-1)$ fixed and forward-cyclically permuting every $n$ consecutive ...
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### Unpublished result of Rosser in Sieve Methods book

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. ...
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### Residues of consecutive primes modulo a fixed integer

It is well-known that the primes are uniformly distributed in residue classes modulo any fixed integer. More precisely, for each integer $q$ and each residue $a \in \mathbb{Z}/q\mathbb{Z}$ that is ...
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### Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number

Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
Zsigmondy's Theorem states that if $a>b>0$ are coprime integers then for any integer $n\geq 1$ there is a prime $p$ that divides $a^n-b^n$ and does not divide $a^k-b^k$ for any positive integer \$...