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 ...

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11
votes
1answer
465 views

Is there a known primitive recursive upper bound on the nth “Zhang prime”

(This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.) In 2013 Zhang showed that there are infinitely many pairs of primes which are less that ...
3
votes
1answer
329 views

Have there been any new developments in the Firoozbakht conjecture? [duplicate]

Let $(p_{n})_{n∈ℕ}$ be the sequence of consecutive primes. In P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1995, page 185, the author says: A new conjecture by F. ...
-1
votes
0answers
35 views

Small subsets of numbers with/without large factors [closed]

Fix $a,b,c\in [\epsilon,\frac{1}{2}-\epsilon]$ for some $\epsilon\in(0,\frac{1}{2})$ with $a<b$. For every $k\in\Bbb N$, fix $i_{k,1},\dots,i_{k,j_k}$ with ...
4
votes
0answers
125 views

The divisors of $p-1$ and high-degree residues modulo $p$

Here is a somewhat more explicit version of a question that I asked a while ago. Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of ...
0
votes
0answers
66 views

Lucasian Primality Criterion for Specific Class of $k \cdot 2^n-1$

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 nonnegative integers . Conjecture Let $N=k\cdot 2^n-1$ such ...
2
votes
0answers
90 views

counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer. Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...
4
votes
1answer
169 views

References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
2
votes
0answers
102 views

Siegel Walfisz Theorem for algebraic number fields

Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.
0
votes
0answers
97 views

E- and A-algorithms for finite arithmetic prime progressions and other sets

There is certain Eratosthenes spirit to my problem (See below). First of all I'd like to stress the mathematical aspect of my question. Also, my question does not amount to the divide and conquer ...
3
votes
1answer
250 views

Sum of two squares and implication of Bunyakovsky conjecture

Bunyakovsky conjecture states that a polynomial with integer coefficients takes infinitely many prime values at integers, unless this is impossible for trivial reasons. Let $a_1(x), a_2(x), a_3(x), ...
1
vote
1answer
182 views

every arithmetic progression contains a sequence of $k$ “consecutive” primes for possibly all natural numbers $k$?

I ask the same question here:http://math.stackexchange.com/q/1019404/192097 writing a little better the previous question: it´s true that if we let $a$ and $b$ be coprime integers, then the ...
5
votes
1answer
386 views

Are there infinitely many primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity?

This is a subsequence of primes $p$ for which $p^2-1$ has at most 6 prime divisors counted with multiplicity. This sequence described in the question is the sequence A079153 in OEIS. I could not ...
6
votes
0answers
155 views

Is ${\rm lcm}\{{\rm ord}_p(q)\colon q\mid p-1,\ q>2\}>\sqrt p\ \,$?

The following question is "ideologically related" to the one I recently asked here. For a prime $p$, let $M_p$ denotes the least common multiple of the orders modulo $p$ of all odd prime divisors of ...
-2
votes
1answer
90 views

Looking for the name of an infinite sequence [closed]

I am looking for information about a sequence that seems like it should converge. The sequence is textually described as: ...
6
votes
0answers
185 views

A conjecture of Erdos on consecutive differences of primes

Let $d_k = p_{k + 1} - p_k$ be the difference between consecutive primes and define \begin{equation} e_k = \left\{\begin{array}{c l} 1 &, d_{k + 1} > d_k \\ 0 &, \text{otherwise} ...
0
votes
0answers
120 views

Estimating the number of twin primes of given natural configuration order

This question is a follow-up from About Goldbach's conjecture. Let $$\mathrm{Co}_{k}(x):=\{n\le x:\mathrm{ord}_{c}(n):=\pi(\sqrt{2n-3})=k\},~~~\mathrm{co}_{k}(x):=\vert\mathrm{Co}_{k}(x)\vert$$ ...
0
votes
0answers
95 views

Lower bound on number of smooth values of polynomial at primes

Given a polynomial $f$, it is known believed that the number of smooth values of $f$ has a positive proportion (for fixed $u$, $\lim_{X\rightarrow\infty} \frac{|\{ n < X\ :\ f(n)\ is\ X^u\ smooth ...
10
votes
1answer
391 views

Parity of the Prime Counting Function

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers. Let: $\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 ...
11
votes
0answers
207 views

Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?

For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$: $$ F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}; $$ thus, for instance, ...
0
votes
1answer
241 views

Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,…$

This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers: $2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$ The $n^{th}$ ...
3
votes
0answers
133 views

Effective version of the Bombieri-Vinogradov theorem

Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?
6
votes
1answer
617 views
+50

What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
3
votes
0answers
526 views

Second Hardy-Littlewood Conjecture theme

If Second Hardy-Littlewood Conjecture is true then we can claim that $\pi(x)-\pi(y)\leq \pi(x-y)$. Thus the conjecture gives an upper bound for the number of primes between $x$ and $y$. I have found ...
5
votes
1answer
266 views

Parity of primes [duplicate]

While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...
5
votes
1answer
265 views

What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$: $\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...
2
votes
0answers
98 views

The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...
7
votes
0answers
215 views

On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
1
vote
0answers
73 views

Definition of primitive divisor of a Lucas sequence

If $\{a_n\}_{n=1}^\infty$ is a sequence of integers, then the definition of primitive divisor of one of its terms is quite natural: Def. 1. A prime number $p$ is a primitive divisor of $a_n$ if $p ...
3
votes
1answer
495 views

what would be the consequences on the distribution of primes of $\Lambda=\infty$?

It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...
2
votes
3answers
278 views

Prime factors of the members of a certain recurrence

It is possible to prove elementarily that there are infinitely many primes that divide some element of the sequence $a_0 = k\ge 0$, $a_n = a_{n-1}^2+ 1$ for all $n\ge 1$ by showing that for all $m$, ...
0
votes
1answer
306 views

The periodic architecture underlying the natural numbers [closed]

EDIT In the original version of this post I did not include a well specified mathematical question, and learning by failing, I realize there should have been. Closing or deleting the question is the ...
15
votes
0answers
386 views

Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is: Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that $$ \binom{2n}{n} \equiv 2\pmod p ? $$ ...
1
vote
1answer
234 views

Conjecture on prime numbers

Given a prime $p$, let $a_n=pn+n-1$. I have noticed that $\forall{p}\exists{n}\in[2,p]:a_n\in\mathbb{P}$. For example: $p=7,a_3=23,a_4=31,a_6=47$. What is this conjecture called, and has it been ...
-3
votes
1answer
174 views

Andrica's and Legendre's Conjectures [closed]

My question is, which of these two conjectures is stronger, Andrica's or Legendre's? Could proving one prove the other? If the upper bound for the prime gap above any given natural number $n$ were to ...
3
votes
1answer
177 views

Prime divisors of the respectively minimal binomial coefficients

In view of Chebyshev's approach to prime numbers, I would like to ask about the regularities and peculiarities of the two sequences $\ \beta(n)\ $ and $\ \gamma(n),\ $ which I define as follows: $\ ...
2
votes
2answers
323 views

Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function. Numerical evidence suggests these identities: $$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...
2
votes
0answers
76 views

Bounds re Asymptotic Formula for the Sum of Largest Prime Factors

I have a reference request related to the result : $\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$ where $P(n)$ is the largest prime factor of the positive ...
0
votes
0answers
185 views

Approximating the number of primes

Let $\pi(x)$ be the number of primes le than $x$. Is there any approach or attempt to find a nice function $f$ that does a better job in approximating $\pi$ than the logarithmic integral ...
2
votes
2answers
383 views

Is there a von Koch-type theorem for the generalized Riemann hypothesis?

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound $$ \mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x). $$ Q1: ...
31
votes
2answers
985 views

Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$. For example $f(6)=6+3+2=11$, $f(5)=5$. Note that $x$ is a fixed point for ...
5
votes
0answers
229 views

$x^2+1$ attaining almost prime values

Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
0
votes
0answers
66 views

Metric defined over Galois extensions of the rationals [duplicate]

I don't know if this of interest, but I'd be curious to know if the following idea has been pursued. In this question (Metric on the set of subsets of the rational primes) I proposed a metric, d, ...
8
votes
2answers
592 views

Is a Galois extension of the rationals determined by its set of completely split primes?

apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the ...
3
votes
1answer
240 views

Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
3
votes
0answers
313 views

Metric on the set of subsets of the rational primes

Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version. I was thinking how to say that two sets ...
9
votes
1answer
257 views

Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$: $$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$ And here is a graphical representation of the 16-digit "repetend," as a directed ...
3
votes
1answer
346 views

Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t. $$f(x)\ll |\psi(x) - x|$$ where $\psi$ is the Chebyshev function.
3
votes
1answer
385 views

Smallest prime in an arithmetic progression

Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...
8
votes
1answer
228 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 ...
7
votes
1answer
240 views

lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?

Are there known any lower and upper bounds for $$ \sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k, $$ where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$? Or at least is it known ...