**4**

votes

**1**answer

276 views

### Upper bound for the first Hardy-Littlewood conjecture

About the Hardy-Littlewood conjecture by Terence Tao:
Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed ...

**2**

votes

**0**answers

298 views

### How big can a set of integers be if all pairs have bounded gcd

In this recent MO question, it was shown that the maximal cardinality of a subset $A(M,N)$ of $[1,N]$ where the pairwise GCD's of all set elements are upper bounded by $M,$ with $M^2\leq N$ has size ...

**4**

votes

**1**answer

390 views

### About factorization in Zhang's proof of weak Twin Prime conjecture

Why does it need to firstly factorize the number n into two factors q and r( Lemma 4 in the paper,see the following)? What's the motivation. What if it doesn't do this factorization?

**0**

votes

**2**answers

305 views

### Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...

**2**

votes

**0**answers

131 views

### On sets of coprime numbers

We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by ...

**2**

votes

**1**answer

148 views

### Gap between semiprimes

Is there a conjectured gap between semiprimes?
There is a conjectured gap between primes in form of Cramer's conjecture. Using this we have $p_1\leq p_0+c(\log p_0)^2$ for consecutive primes $p_0$ ...

**22**

votes

**1**answer

650 views

### How big can a set of integers be if all pairs have small gcd?

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?
If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ ...

**6**

votes

**1**answer

186 views

### $N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

Assume that $ab \neq 0$. What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could ...

**8**

votes

**0**answers

161 views

### Primes of the form $(2m+1)^2-2^{2s+1}$

The question is the following :
Question:
Does there exist infinitely many primes of the form $(2m+1)^2-2^{2s+1}$ with $m,s\geq 1$ ?
Why this could be true:
Bunyakowsky conjecture would ...

**2**

votes

**0**answers

195 views

### Avoiding Chinese Remainder Theorem

Given $k\in\Bbb N$ with $k<(\log_2N)^{\frac1\alpha}$ where $\alpha>2$ is fixed and $N$ being some integer such that $$N<\prod_{i=1}^k\pi_i^{a_i}$$ where $\pi_1,\pi_2,\dots,\pi_{k-1},\pi_k$ ...

**0**

votes

**0**answers

108 views

### Bounding $\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}$ with $1\ll r(x)\ll \log^{4}(x)$

I would like to know whether it is possible to obtain the bounds $\sqrt{r(x)}\ll k(x)\ll r(x)$ where $k(x)={\pi(x+r(x))-\pi(x-r(x))}$ and $1\ll r(x)\ll \log^{4}(x)$ and thus ...

**2**

votes

**2**answers

340 views

### Primes $p$ for which $2p-1$ is prime

It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?
Seemingly it's also an open problem (see here and the linked ...

**3**

votes

**1**answer

178 views

### Is $2^n -1$ finitely many times the product of consecutive primes? [duplicate]

This question was asked at MSE but recieved no attention at all.
Here it is:
Are there finitely many $(n,k) \in \mathbb{N}^2$ with $2^n-1=p_1p_2\cdots p_k$ ?
$p_1=3,p_2=5 , ...,p_k$ are ...

**4**

votes

**0**answers

97 views

### Behavior of the “mean prime factor” of numbers

This question concerns the behavior of
a function $f(\;)$ that maps each number in $\mathbb{N}$ to
its mean prime factor.
I previously posted premature questions, now deleted, which
explains the cites ...

**8**

votes

**1**answer

313 views

### Distribution of the number of prime factors

Count the number of prime factors of a number $n$
to include multiplicity,
so that
$$n=24=2^3 \cdot 3 = 2 \cdot 2 \cdot 2 \cdot 3$$
has $4$ prime factors, and
$$n =
6500 =
2^2 \cdot 5^3 \cdot 13 =
2 ...

**3**

votes

**1**answer

277 views

### Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least ...

**1**

vote

**0**answers

120 views

### A Question on Chinese Remainder Theorem [closed]

Let $p_1,p_2,\ldots,p_n$ be odd primes.
Let $y$ be the unique solution of the Chinese Remainder Problem ( $0 \le y < m$, $ m = p_1\cdot p_2 \cdots p_n$)
$ y = \frac{p_1-1}{2} \text{ mod } (p_1) ...

**7**

votes

**3**answers

519 views

### Quantitative and elementary proofs of the Prime Number Theorem

I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...

**6**

votes

**0**answers

198 views

### Any ways to Simplify Daboussi's Argument for Prime Number Theorem?

One strategy to prove the Prime number theorem involves removing some factors:
$$ \limsup_{x \to \infty} \underbrace{\frac{1}{x}\sum_{n \leq x} \mu(x)}_{\color{red}{A}}\leq
\prod_{p \leq y} \left( 1 ...

**4**

votes

**1**answer

470 views

### Green-Tao theorem for 1-central numbers

This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the ...

**18**

votes

**2**answers

3k views

### Does the equation $241+2^{2s+1}=m^2$ have a solution?

Let $p$ be a prime congruent to $1$ mod. 8.
If $p= 17$ one has : $p+ 8 = 5 ^2$.
If $p= 41$ one has : $p+ 8 = 7 ^2$.
If $p= 73$ one has : $p+ 8 = 9 ^2$.
If $p= 89$ one has : $p+ 32 = 11 ^2$.
If ...

**4**

votes

**2**answers

276 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

597 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

457 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

118 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

65 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 ...

**22**

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

332 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

891 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

442 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 ...

**2**

votes

**2**answers

266 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

97 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

300 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

144 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

185 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

175 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

387 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

187 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

305 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

270 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 $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$
...

**14**

votes

**2**answers

384 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

353 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

138 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

254 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 $k$ ...

**3**

votes

**4**answers

327 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

414 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

101 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

364 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 $10^m+1,10^m+3,10^m+7$ ...

**7**

votes

**0**answers

291 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 ...