# Tagged Questions

**8**

votes

**1**answer

209 views

### Integers with a large prime divisor in short intervals

For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers ...

**-1**

votes

**0**answers

55 views

### Property of set of prime numbers [migrated]

let $\{p_1,p_2,p_2,\cdots ,p_r\}$ be the set of $r$($\ge2$) pair wise distinct prime numbers
i.e.., $(i\ne j \implies p_i \ne p_j)$ for all $1\le i,j\le r$
${Statement}$ :
For any such ...

**-1**

votes

**1**answer

131 views

### Conjectured Primality Test for Numbers of the Form k2^n+1 with n>2 [closed]

Definition : 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) $
where $m$ and $x$ are positive integers .
Conjecture : Let $N=k\cdot 2^n+1$ with ...

**2**

votes

**0**answers

81 views

### Primality Criterion for Specific Class of Numbers of the Form kb^n-1

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ ,
$k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ .
Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...

**2**

votes

**0**answers

86 views

### Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$
Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where
...

**7**

votes

**0**answers

158 views

### In which orders can the numbers of prime factors of consecutive integers be?

Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given ...

**2**

votes

**0**answers

122 views

### n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...

**2**

votes

**1**answer

327 views

### Looking for paper: Weil's original 1952 “Sur les formules explicites de la théorie des nombres premiers”

I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...

**7**

votes

**1**answer

311 views

### Are primes of density 0 in $a\cdot b^n+c$?

Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms ...

**6**

votes

**1**answer

257 views

### Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?

(I've taken this from MSE, it seems to be more appropriate here)
I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the
Question for
$$ b^{p-1} \equiv 1 \pmod{ ...

**10**

votes

**1**answer

568 views

### Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum?
$$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{n}} ...

**0**

votes

**0**answers

214 views

### On $n$-th prime $\pmod {n}$

Has it been proved or disproved that for any fixed $a\geq 1$ there are infinitelly many primes $p_n\equiv a\pmod{n}$?
I believe i have proved that for every $a\geq1$ there are infinitelly many ...

**6**

votes

**0**answers

228 views

### Generalizing prime numbers to product-indecomposable objects in toposes

Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of ...

**7**

votes

**1**answer

450 views

### The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
...

**3**

votes

**1**answer

256 views

### Primes in short intervals with a preassigned frobenius

Edited after mistake in the first version.
It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a ...

**4**

votes

**1**answer

288 views

### A “bit” of primes

Is there anything known/proved/conjectured about the distribution of:
$$B(n) = \frac{(p_n-1)}{2} \bmod 2, \qquad p_n \mbox{ is the } n\mbox{-th prime}$$
i.e. the bit 1 of the binary representation ...

**5**

votes

**2**answers

294 views

### Equivalence of two well-known forms of (RH): reference-request.

This is a reference-request about a very simple statement.
The Riemann hypothesis is well-known to be equivalent to
$$(1)\ \ \ \pi(x) = \mathrm{Li}(x)+O(x^{1/2} \log x)$$
and to
$$(2)\ \ ...

**5**

votes

**1**answer

226 views

### Large gaps between P2s

Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...

**3**

votes

**3**answers

452 views

### Estimate for products of integers that are relatively prime with $N$

Let $N$ be a positive integer. Are there known estimates for the product of all numbers that are smaller than $N$ and relatively prime with $N$? One can assume that $N$ is free of squares, if this ...

**1**

vote

**0**answers

221 views

### Reference request: Dickman, On the frequency of numbers containing prime factors …

I've been trying without success to find the paper
Dickman, Karl, "On the frequency of numbers containing prime factors of a certain relative magnitude." Ark Mal., Astronomi och Physik, 22A (10), ...

**2**

votes

**3**answers

603 views

### Proof of infinitude of primes whose reversal in base 10 is also prime [closed]

Is there any proof of infinitude of A007500 primes?
If you want to generate them here is trivial and naive python program.
...

**2**

votes

**1**answer

530 views

### Primes and Ackermann's function

If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all?
EDIT:
I ...

**15**

votes

**2**answers

1k views

### Primes of the form $x^2+ny^2$ and congruences.

The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...

**5**

votes

**2**answers

571 views

### ASCII prime plots and prime-rich quadratic polynomials

This is a series of questions inspired by the MathOverflow question
Find the least prime so that p-1 has two factors greater than $m$ and $n$ posted by Aaron Sterling.
I suggested plotting primes by ...

**1**

vote

**1**answer

416 views

### Are large numbers the sum of two or more large primes? [Hoping for reasonable constants]

Is it true that for all $n>N$ that n is the sum of two or more distinct primes that are either large or (for parity reasons) 2?
I feel like I've seen a result allowing this with $p\gg n^e$ for ...

**3**

votes

**0**answers

489 views

### Paul Erdős and Ramanujan Primes

It's easy to find Ramanujan's proof of Ramanujan primes:
Ramanujan's Proof
Wikipedia mentions that Paul Erdős also had a proof:
Wikipedia article on Bertrand's Postulate
Does anyone know the ...

**2**

votes

**0**answers

271 views

### Reducing factoring prime products to factoring integer products (in average-case)

My question is about the equivalence of the security of various candidate one-way functions that can be constructed based on the hardness of factoring. (This question has been asked also in the CS ...

**3**

votes

**2**answers

490 views

### Asymptotics for the number of partitions of $n$ into odd prime parts

Hello!
I am interested in the asymptotic behavior of the function $p_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - http://oeis.org/A099773 .
I couldn't find any ...

**12**

votes

**3**answers

2k views

### The multiplicative order of 2 modulo primes

Artin's Conjecture says that any positive integer, which is not a square, is a primitive root modulo infinitely many primes. Christopher Hooley gave in
Hooley, Christopher (1967). "On Artin's ...

**1**

vote

**0**answers

198 views

### Pierpont primes

A Pierpont prime is a prime $p$ that can be written as $$p=2^u 3^v + 1.$$
What is known about Pierpont primes? I'm not a number theorist, and the best I can find is
...

**6**

votes

**4**answers

667 views

### Reference for the expected number of prime factors of n larger than n^alpha is -log alpha

Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.
It's my understanding that ...

**14**

votes

**3**answers

2k views

### Twin Prime Conjecture Reference

I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...