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 ...

**-3**

votes

**0**answers

51 views

### Number theory, discrete mathematics [on hold]

Let m>=2 be a prime number. Let x be any integer that is not congruent to 0(mod m). Show that there is a unique integer y which is less than m such that xy is congruent to 1(mod m).
I've worked out a ...

**2**

votes

**1**answer

131 views

### The largest number $y$ such that $(x!)^{x+y}|(x^2)!$

Since the multiplication of $n$ consecutive integers is divided by $n!$, then $(n!)^n|(n^2)!$ with $n$ is a positive integer.
Are there any formula of the function $y=f(x)$ that shows the largest ...

**7**

votes

**1**answer

154 views

### For which primes does this iterated function act transitively? (Sort of a finite analogue of Collatz conjecture.)

Background: I was trying to prove something having to do with cyclic group actions on matroids and was able to show that what I want holds if a particular elementary-looking number-theoretic property ...

**1**

vote

**1**answer

579 views

### Is every prime greater than 5, less than the sum of the two previous primes?

Can one prove by elementary means (such as Paul Erdös' proof of Bertrand's Postulate) that every prime greater than 5 is less than the sum of the two primes immediately preceding it?

**5**

votes

**1**answer

288 views

### consecutive prime gaps and explicit bound

I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...

**6**

votes

**1**answer

196 views

### Prime plus square equals prime

Furstenberg–Sárközy's theorem states that if a set of positive integers has positive upper density, then there exists (infinitely many) pair of elements of the set, whose difference is a perfect ...

**6**

votes

**0**answers

100 views

### Does the primality of the number of dimensions affect its properties?

Motivated by How does the parity of a dimension affect its properties? I dare to ask the following question (with thanks to my colleague Vedran Dunjko): We happen to live in a world of prime ...

**1**

vote

**1**answer

74 views

### How many iterations the best biprime factoring method has to factor a number [closed]

I'm researching method of biprime number factoring. I have a biprime number 1012322327 * 1115382761 (19 decimal digits= 1129126872111204847). I'd like to know how many iterations (or trials) the best ...

**-1**

votes

**0**answers

61 views

### $ \alpha $ s.t. $ (1-\varepsilon)\log^{\alpha}p_{n}\lt g_{n}\lt(1+\varepsilon)\log^{\alpha}p_n $ is dense in some open interval centered on $1$

Say a real $ \alpha $ such that the proportion of $ n $ such that $ (1-\varepsilon)\log^{\alpha}p_{n}\lt p_{n+1}-p_{n}\lt(1+\varepsilon)\log^{\alpha}p_{n} $ is a positive increasing function of $0\...

**12**

votes

**3**answers

727 views

### Stronger versions of Wilson's Theorem

Problem
Let $c \in \mathbb{N}$ $;$ $\exists$ a prime $p$ for which:
$$p^c \mid (p-1)!+1$$
Does $\exists$ $M$ $\in$ $\mathbb{N}$ $;$ $\forall$ $c \geqslant M$
$;$ $\nexists$ $p$ ...

**0**

votes

**0**answers

233 views

### Conjectured primality tests for specific classes of $k\cdot b^n \pm 1$

Can you provide proofs or counterexamples for the claims given below?
Inspired by Lucas-Lehmer-Riesel primality test I have formulated the following two claims:
First claim
Let $P_m(x)=2^{-m}\...

**3**

votes

**1**answer

151 views

### Almost-Primes in Short Intervals

Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number ...

**2**

votes

**0**answers

171 views

### What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?

What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...

**1**

vote

**1**answer

309 views

### Primality test for generalized Fermat numbers

This question is successor of Primality test for specific class of generalized Fermat numbers .
Can you provide a proof or a counterexample for the claim given below?
Inspired by Lucas–Lehmer–Riesel ...

**7**

votes

**1**answer

258 views

### Riemann sum formula for definite integral using prime numbers

I had asked this question in MSE. It got lot of upvotes but no answer (except one which was too long to be posted as a comment) hence I am posting it in MO.
While answering another question in MSE I ...

**0**

votes

**0**answers

104 views

### What is known about the lower bound for the integers $n$ for which $n$ minus the first $k$ odd primes are $k$ composite numbers?

Question edited in view of the comments below
By Yamada's paper we can conclude that if $n>e^{e^{36}}$ be an even number then it can always be written as the sum of a prime and a semi-prime.
My ...

**0**

votes

**0**answers

52 views

### Distribution of witnesses that yields a factorization of $n$

For each natural odd number $n$, let $A_n$ be the set of all $1 < a < n$ for which $n$ is a pseudoprime to base $a$ but not a strong pseudoprime to base $a$. This is the set of bases for which ...

**12**

votes

**1**answer

341 views

### Is the identity function a unique multiplicative homeomorphism of $\mathbb N$?

Endow the set $\mathbb N$ of positive integers with the topology $\tau$ generated by the base consisting of arithmetic progressions $a+b\mathbb N_0$ where $\mathbb N_0=\{0\}\cup\mathbb N$, where $a,b\...

**6**

votes

**2**answers

267 views

### A simultaneous generalization of the Grunwald-Wang and Dirichlet Theorems on primes

By Grunwald-Wang Theorem, if for some odd number $n$ the equation $x^n=a$ has no solutions in $\mathbb Z$, then the equation $x^n=a\mod p$ has no solutions for some prime number $p$. I am interested ...

**7**

votes

**2**answers

728 views

### A stronger form of the Dirichlet Theorem on prime numbers in arithmetic sequences

Question 1. Let $a,b>1$ be two natural numbers. Is there a prime number $p\in 1+b\mathbb N$ such that $a+p\mathbb Z$ is a generator of the multiplicative group of the field $\mathbb Z/p\mathbb Z$?
...

**0**

votes

**0**answers

122 views

### Counting divisors of primes plus or minus 2

I was naively exploring the (usually) composites $\pm 2$ from a
prime $p$, wondering if there might be some asymmetry,
and made this histogram of the difference $\Delta$ in
the number of divisors of $...

**3**

votes

**1**answer

228 views

### Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambda(m)=\...

**3**

votes

**1**answer

271 views

### Primality test for specific class of $N=k \cdot b^n-1$

This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ .
Can you provide a proof or a counterexample to the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(...

**1**

vote

**0**answers

85 views

### 1-concatenable primes

If we choose some prime, say $11$, we can concatenate one digit to the left and one to the right to obtain another prime, for example, $2113$, we can do the same with $2113$ and obtain $121139$, a ...

**11**

votes

**0**answers

285 views

### Prime numbers of the form $1+p+p^2+\dots+p^n$

I am looking at prime numbers of the form $Q=1+p+p^2+\dots+p^n$, where $p$ is also a prime number, $n$ is finite. Unfortunately I was unable to find any reference to such prime numbers in the ...

**0**

votes

**0**answers

82 views

### How to prove a Mersenne number is not pseudoprime to the base 3?

I start think in finding a small divisor $p$ of $M_n$ then I obtain that $p\mid (3^{M_{n}-1}-1)$ hence $ord_{p}(3)\mid M_{n}-1$
Now can I prove p is the same as $M_{n}$ so that $M_{n}$ is not ...

**-1**

votes

**1**answer

126 views

### The power of a prime in the prime factorization of a factorial [closed]

How do we find—for example—how many $5$s are in the prime factorization of $n!$? I've read that it is $\lfloor n/5 \rfloor$, but why is that?

**0**

votes

**0**answers

62 views

### Decomposition of prime into $q \cdot r + s$, where $q,r,s$ are primes

Moments ago I created a following conjecture:
If $p$ is prime different from $2,3,5,7$ then there exist primes $q,r,s$ such that $q<p$ and $r<p$ and $s<p$ and $p=q \cdot r +s$.
Is this true?...

**8**

votes

**1**answer

855 views

### Are there infinitely many primes of this form?

The semiprime $87 = 3*29$ has a curious property: it's the fact that both
$87^2 + 29^2 + 3^2 = 8419$
and
$87^2 - 29^2 - 3^2 = 6719$
are prime numbers.
This intrigued me and led me to wonder if ...

**3**

votes

**1**answer

246 views

### A specific Diophantine equation restricted to prime values of variables.

Consider the following Diophantine equation $$x^2+x+1=(a^2+a+1)(b^2+b+1)(c^2+c+1).$$ Assume also that $x,a,b,c,a^2+a+1,b^2+b+1, c^2+c+1$ are all primes. We'll call such a quadruplet $(x,a,b,c)$ a ...

**0**

votes

**0**answers

120 views

### An elementary size bound in number theory?

Given integer $a$ of size $R^\alpha$ with $\alpha\in(0,1)$ and $t$ large primes $R_i$ of roughly same size $R$ what is the smallest $T$ one needs so that there is an integer $0<K<R$ with as ...

**4**

votes

**1**answer

135 views

### Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$.
What is the best method to find all such ...

**5**

votes

**1**answer

310 views

### Does every prime $p$ appear in a $p$-term arithmetic progression of primes? [duplicate]

This is a follow-up to an earlier question.
The answer to that question was found on this page. The discussion on OEIS seems to suggest that, for any prime $p$, there should exist a $p$-length ...

**29**

votes

**4**answers

3k views

### Is there an 11-term arithmetic progression of primes beginning with 11?

i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?

**12**

votes

**1**answer

260 views

### Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$

Let $p=8k+1\equiv 1\pmod 8$ be a prime, thus $2$ is a quadratic residue module $p$. Euler's criterion show that $$2^{\frac{p-1}{2}}\equiv 1 \pmod p.$$
So we must have
$$2^{\frac{p-1}{4}}\equiv \...

**7**

votes

**2**answers

397 views

### Primality test for specific class of Proth numbers

Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
Let $N=k\cdot 2^n+1$...

**-3**

votes

**1**answer

218 views

### Is it true that there are infinite palindromic primes that when squared give palindromic number? [closed]

Can you prove that there are infinite palindromic primes that when squared give a palindromic number?

**6**

votes

**0**answers

118 views

### $p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?

Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I came ...

**8**

votes

**1**answer

324 views

### Goldbach's conjecture for the Liouville function

Is it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ?
Here $\lambda$ is the Liouville function.

**0**

votes

**0**answers

97 views

### Does this recurrence yields only prime numbers?

The title of this question is merely illustrative, since I do not expect an answer by yes or not, but I seek some reference on the subject.
In this article: https://pdfs.semanticscholar.org/e9be/...

**4**

votes

**1**answer

322 views

### Density of integers with a large rough divisor

Let $N < 2^a$ be a positive integer chosen uniformly at random. Let $\tilde{N}$ be the result of removing from $N$ all its prime factors less than $2^b$. What is the probability that $\tilde{N}$ is ...

**6**

votes

**1**answer

187 views

### Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?

Let $p$ be an odd prime, $g$ a primitive root of $p$ and $A=\{1, 2, \ldots, p-1 \}$.
Obviously, $\sigma_g(p)=\begin{pmatrix}
1 & 2 & \ldots & {p-1} \\
g^1\pmod{p} & g^2\pmod{p} &...

**10**

votes

**1**answer

469 views

### How many zeta zeros are needed to accurately calculate five digits for π(1000000), where π(x) is the prime counting function?

John Derbyshire in his book PRIME OBSESSION says on page 343:
"I’ll round off with a complete calculation of $\pi(1,000,000)$, the
number of primes up to one million, using Riemann’s formula -- ...

**6**

votes

**1**answer

230 views

### Upperbounding a sum of Legendre-Symbols

Let $p$ be a prime with $p\equiv 3 \mod 4$, for any $\mathcal{I} \subset \lbrace 0,...,p-1 \rbrace $ and any $\mathcal{J} \subset \lbrace 0,...,p-1 \rbrace $ with $\vert\mathcal{I}\vert \leq \sqrt{p} $...

**4**

votes

**1**answer

389 views

### Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$

Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the ...

**2**

votes

**2**answers

150 views

### Sum of a terms in a divergent series taken along indices the sum of whose reciprocal diverges. Can the sum converge?

Let $\{x_n\}_{n=1}^{\infty}$ be a monotone decreasing sequence of positive real numbers such that $\sum_{n=1}^{\infty} x_n$ diverges. Also let $\{k_n\}_{n=1}^{\infty}$ be a strictly increasing ...

**10**

votes

**1**answer

356 views

### Spacing of fractions with prime denominator

Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ ...

**4**

votes

**1**answer

357 views

### Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?

Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.
I stumbled ...

**3**

votes

**0**answers

126 views

### Cancellation in this exponential sum?

I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.

**0**

votes

**1**answer

98 views

### Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?

I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:
For $n, x \ge 2$ be two integers then:
$$P_{2n} \ge 2P_n$$
and
$$\pi(2x) \le 2\pi(x)...