**4**

votes

**2**answers

291 views

### Relative-totient function (2nd attempt)

Let $\Lambda(x,y)$ be the count of totatives of $x$ that are less than or equal to $y$.
I am asking for the following result to be verified, (particularly the final proposal), I have found no ...

**-1**

votes

**1**answer

603 views

### What is wrong with this counterexample to primality test assuming GRH? [closed]

From SMOOTH NUMBERS: COMPUTATIONAL NUMBER THEORY AND BEYOND Andrew Granville pp.13-14:
2j. Lenstra’s polynomial time test as to whether an integer that is conjecturally prime, is rigorously ...

**6**

votes

**1**answer

484 views

### An elementary lower bound on the number of primes

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$.
In a hunt for an ...

**1**

vote

**1**answer

124 views

### A deterministic and explicitly described walk which is like random ones

Consider a sequence $(X_i)_{i = 1}^{\infty}$ which every $X_i$ is $-1$, $0$ or $+1$ and lets define $Y_n = X_1+ \cdots + X_n$. We say the sequence $(X_i)_{i = 1}^{\infty}$ a Good Sequence if $Y_n \neq ...

**0**

votes

**0**answers

68 views

### Is this sufficient condition for primality of numbers of special form of practical interest?

The best variant of deterministic primality is of complexity $ \tilde{O}(\log^{6}(n))$.
For large $n$, this is significantly worse than $O(\log{n})$.
Appears to me paper p. 16
gives sufficient ...

**23**

votes

**2**answers

1k views

### Proof for new deterministic primality test

Claim:
Let $p$ be a positive prime. Let $n \in \left\{1, 2, 3, ...\right\}$. Then $N =
p\cdot 2^n+1$ is prime, if and only if it holds the congruence $3^{(N-1)/2} \equiv \pm1\ ($mod $N)$.
If the ...

**1**

vote

**1**answer

338 views

### Proof that $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log {p_n}$ [closed]

I cannot find a proof of this theorem. May anyone assist?
$p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log{p_n}$

**10**

votes

**2**answers

911 views

### Update for 2015: least prime of form nq+1, with q prime?

I have received a complaint about my 2011 answer
least prime in a arithmetic progression
which, indeed, gives conflicting reports about this:
given a prime $q,$ what can we say about an upper ...

**7**

votes

**2**answers

454 views

### What is wrong with this deterministic algorithm efficiently generating large primes?

According to PolyMath
(Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in ...

**3**

votes

**2**answers

295 views

### Asymptotics of the least common multiple of the first natural numbers

What is $$ \limsup_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \ \ ?$$

**3**

votes

**1**answer

100 views

### Upper bound for OEIS A076689 “Smallest k such that k*p#+1 is prime”?

OEIS A076689
Is defined as smallest integer $a(n)=k$ such that $k n\#+1$ is prime,
where $n\#$ is primorial, the product of the first $n$ primes.
Lower bound appears $1$, the primorial primes.
...

**-3**

votes

**2**answers

309 views

### The number of totatives to the nth primorial, in an interval shorter than the nth primorial

(The notation of this question will be improved over the next few days, sorry for the lack of clarity at the moment.)
Can, and if so when can, we determine the amount of natural numbers which are ...

**1**

vote

**1**answer

147 views

### Primes in simultaneous arithmetic progressions

Suppose we're given four positive integers $a$, $b$, $c$, $d$ such that $a$ and $b$ are coprime, and $c$ and $d$ are coprime. Is there a non-negative integer $k$ such that both $ak+b$ and $ck+d$ are ...

**-1**

votes

**1**answer

188 views

### Number of different factors of given size in primorial

Let $b_n$ be number of bits in product of all primes from $1$ to $n$ which is approximtely $b_n\approx n$.
What is the approximate number of distinct factors with number of bits ...

**7**

votes

**0**answers

178 views

### Are there an infinite number of twin semiprimes?

A semiprime is a number that is the
product of two (possibly equal) primes.
Define twin semiprimes (my terminology) as two consecutive numbers both semiprimes.
For example, $(57,58)$ are twin ...

**6**

votes

**0**answers

396 views

### Prime gap counts in short intervals

Since it is conjectured that the twin prime count at $n\sim2 C_2\ \frac{n}{\log^2n},$ where $C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} = 0.66016 18158 \dots,$ it follows that the twin prime count ...

**2**

votes

**2**answers

198 views

### Finiteness of number of consecutive primes with gap $4$

Assuming Riemann Hypothesis Hardy showed primes $3\bmod 4$ are more common than primes $1\bmod 4$ https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences_of_the_generalized_Riemann_hypothesis.
...

**6**

votes

**1**answer

308 views

### Negative coefficient in an almost cyclotomic polynomial

Let $a,b,c,d$ be four prime numbers. We set the polynomial :
...

**4**

votes

**0**answers

280 views

### Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ?
Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$
...

**18**

votes

**3**answers

651 views

### Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$.
Probable primes are the union of the primes and base two pseudoprimes.
This definition is much ...

**7**

votes

**1**answer

390 views

### Bounded gaps between primes in arithmetic progressions

Has Zhang's work on bounded gaps between primes been extended to the following theorem?
For any arithmetic progression $an+b,\gcd(a,b)=1$, there is a constant $H$ (depending only on $a$) such that ...

**0**

votes

**0**answers

144 views

### The maximum lengthed sequence of prime numbers with certain conditions (denizens)

Definition - Denizen
A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition;
...

**1**

vote

**1**answer

36 views

### Prime constant graphicial representation [closed]

I have something to check. It is about prime constant (I don't know if it is officially so called), but it is created on following way. We start with binary point number represenation. Zero followed ...

**-5**

votes

**1**answer

281 views

### Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...

**4**

votes

**4**answers

351 views

### Prime divisors of values of a polynomial on an infinite set

This may be a well known problem:
Let $f$ be a polynomial with integer coefficients. Is it possible for the set of prime divisors of values of $f$ on an infinite set of integers, to be finite?
I ...

**5**

votes

**1**answer

418 views

### Unexpectedly prime rich cubic polynomial

We got a cubic polynomial which is unexpectedly prime rich.
Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and
$\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$.
Let ...

**0**

votes

**0**answers

112 views

### Polynomial identities for congruent numbers and Bunyakovsky's conjecture

Bunyakovsky's conjecture states that polynomial with integer coefficients
takes infinitely many prime values unless there are obvious reasons not
to.
It appears to imply something about polynomial ...

**10**

votes

**0**answers

371 views

### Between Fermat's primes and the twin primes

Let me start with a curiosity. The integers $11,13,17,19$ are prime numbers, and $101,103,107,109$ are prime as well. One might wonder whether there is another occurrence where $10^m+1,10^m+3,10^m+7$ ...

**13**

votes

**0**answers

421 views

### Intersection between the sums of the first positive integers, primes and non primes

Is the following conjecture true ?
$$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap
\left\lbrace ...

**1**

vote

**0**answers

97 views

### Can someone explain some of the steps in this paper clearly?

I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$
Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...

**7**

votes

**1**answer

510 views

### Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: ...

**1**

vote

**1**answer

219 views

### reference on Dirichlet theorem on primes in arithmetic progression

I appreciate if you could help me to find a reference (and a proof).
Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...

**0**

votes

**1**answer

252 views

### Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime? [closed]

I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime?
Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of ...

**0**

votes

**0**answers

150 views

### arithmetic progressions with few primes

Is this true ?
Let $\beta_0$ be a positive number. One may find $\beta>\beta_0$, $0<\lambda<1$, and infinitely many $q>1$ so that there exists an arithmetic progression of step $q$, $a_1, ...

**2**

votes

**1**answer

211 views

### Counting function for prime pair with bounded gaps between them [duplicate]

I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible.
Yitang Zhang breakthrough result established that ...

**13**

votes

**1**answer

639 views

### Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $

A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale.
I would like ask about the much weaker statement forgetting ...

**1**

vote

**0**answers

184 views

### Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemann hypothesis they used?

Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemannn hypothesis they used?
In their paper,
Some problems of 'Partitio numerorum'; III - On ...

**7**

votes

**0**answers

567 views

### “Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write:
In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...

**7**

votes

**1**answer

422 views

### Approximating a real by a ratio of primes

Let $x$ and $y$ be positive reals in $(0,1)$ with $x < y$ and $y-x =\epsilon$.
I seek smallest primes $p$ and $q$ such that
$$x \le \frac{p}{q} \le (x+\epsilon) = y \;.$$
Q. What upper bound ...

**3**

votes

**1**answer

217 views

### Least prime for which a square-free integer is a non-residue

Suppose $a$ is a square-free integer and $\left(\frac{a}{p}\right)=1$ for the primes $p\leq k$. I'll call $a$ a quasi-square of order $k$. What I am interested in is the maximum value of $k$ in terms ...

**5**

votes

**0**answers

118 views

### Visibility in a prime orchard

This suggests a variant on Polya's orchard problem.
That problem asks1
for which radius $\epsilon$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the ...

**2**

votes

**2**answers

245 views

### binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result.
Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...

**0**

votes

**1**answer

186 views

### A conjecture on the prime counting function

I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the ...

**5**

votes

**3**answers

417 views

### Logarithmic integral, $π(x)$ and $x/(\ln x)$

The function $\text{Li}$ (logarithmic integral) is defined for $x>0$
by
$$
\text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}.
$$
The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ...

**4**

votes

**1**answer

325 views

### Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...

**2**

votes

**1**answer

405 views

### Number of twin primes

Consider number of twin primes less than $x$. We know that this number less than $\frac{Cx}{\log^2 x}$ for some constant $C$.
Denote by $p_n$ the $n$-th prime number. Do we have the same result ...

**2**

votes

**2**answers

596 views

### If $\binom{2p}{p}$ is $(-1)^{p-1} \bmod 2p+1$ is then $2p+1$ prime?

Let $p$ be a positive integer; if $2p+1$ is prime then it is easily checked that
$$(2p+1)\mid\left(\binom{2p}{p}+(-1)^{p-1}\right);$$
conversely I conjecture that if the above divisibility assumption ...

**5**

votes

**0**answers

175 views

### Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?

Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$.
The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...

**0**

votes

**2**answers

94 views

### Min number of primes up to n

According to the prime number theorem there are about $n/\ln(n)$ primes less than $n$. This value is a limit but it could fluctuate. My question is, is there a known bound on this fluctuation? i.e. ...

**8**

votes

**2**answers

579 views

### Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers

To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes
infinitely many prime numbers as values. Is it known so far whether there is at all any
polynomial $P \in ...