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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1
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1answer
143 views

Is anything like $\phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions?

Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions? Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime ...
7
votes
2answers
474 views

Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid ...
1
vote
0answers
78 views

Best constant for Maier's theorem?

Maier proved that, for fixed $\lambda>1,$ $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1 $$ and in particular $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda ...
9
votes
1answer
1k views

Convergence of the series $\sum_p p^{-s}$ ($p$ prime and $s>1$)

I know that $\sum_p p^{-s}$, $s>1$, converges. Now, I define $J(s) = \sum_p p^{-s}$. Are there any "well known" values for $J(2)$, $J(3)$, $J(4)$, etc? We all know that $\zeta(2)= \frac{\pi^2}{6}$, ...
-3
votes
0answers
62 views

Squares and Prime Numbers [closed]

IF A+BC+CC IS A PERFECT SQUARE. WHERE A,B,C ARE POSITIVE INTEGERS, B>A AND B IS AN EVEN NUMBER. IS IT POSSIBLE TO CALCULATE VALUE OF C IN TERMS OF A AND B?
8
votes
2answers
683 views

What is known about primes of the form x^2-2y^2?

David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
16
votes
1answer
712 views

Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
2
votes
1answer
260 views

$n^3 | \sum_{i=1}^{n-1}\binom{n}{i}^2$ => $n | \sum_{i=1}^{n-1}\binom{n}{i}$?

For $n\in \mathbf{N}$ is $$n^3 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$n \text{ divides } ...
-1
votes
0answers
265 views

Is there a polynomial that generates only primes or semi-primes? [migrated]

I know that no non-constant polynomial function $P(n)$ with integer coefficients exists that evaluates to a prime for every integer value of $n$. My question is - does there exist a non-constant ...
0
votes
0answers
133 views

Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?

For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...
0
votes
2answers
747 views

Yitang Zhang's paper [closed]

I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which ...
0
votes
0answers
81 views

logarithmic integral question

Define: $\operatorname{li}(x)=\int_{0}^{x}\dfrac{1}{\log(t)}\operatorname{d}t$. When does the following statement fail? With $\theta = 1 + \frac{1}{\operatorname{li}(x)}$, for $x \ge x_0$, ...
5
votes
1answer
335 views

Generating primes with floor of a polynomial $[p(n)]$

Is there a polynomial $p(x)$ with real coefitients and degree at least one that $[p(n)]$ for everey natural number like $n$ be a prime? If yes, what is such a polynomial $p(x)$ and if no, how to ...
2
votes
4answers
543 views

solution of the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$

i am wondering if there is a complete solution for the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$ in which $a,b,c,d,k$ are integer(not all zero) and $p$ is odd prime.
10
votes
0answers
281 views

Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
4
votes
2answers
432 views

How to calculate the sum of remainders of N?

I'm trying to sum the remainders when dividing N by numbers from 1 up to N $$\sum_{i = 1}^{N} N \bmod i$$ It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ...
10
votes
1answer
478 views

Divergence of a series similar to $\sum\frac{1}{p}$

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$. We denote the least integer not covered by the arithmetic ...
78
votes
4answers
28k views

Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture

Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville : “The big experts in the field had already tried to make this approach work,” Granville ...
4
votes
0answers
246 views

Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime. ...
2
votes
2answers
399 views

Prime divisors of $p^n+1$

Let $p$ be a rational prime and $n$ be a positive integer. It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or ...
2
votes
2answers
564 views

Has this formula about prime gaps already been conjectured and/or proven?

While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p_{b}-p_{a}}{b-a}$ where $a<b$ are positive integers: ...
1
vote
0answers
423 views

Is $ a^2b^2-ab-ap$ a perfect square for suitable $a,b\in \mathbb{Z}^+$?

Consider the expression $$ a^2b^2-ab-ap\qquad (a,b\in \mathbb{Z}^+), $$ where $p\equiv1\pmod{4}$. Question. For every prime $p\equiv1\pmod{4}$ do there exist $a,b\in\mathbb{Z}^+$ such that ...
2
votes
1answer
232 views

Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states: Unconditionally we have \begin{equation} \pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x ...
2
votes
0answers
357 views

A “Take a Square Root When You Can” conjecture related to the prime factorization

I would tend to think that the following has already been investigated. But as implied from the title, I have no idea how to even start looking for it. Let $P_n$ denote the sum of the squares of ...
2
votes
1answer
496 views

Conjecture on the square root of the sum of the squares of the prime factors of a number

Let $A_{n}$ denote the square root of the sum of the squares of the prime factors of $n$. For example, $A_{60}=\sqrt{2^2+2^2+3^2+5^2}\approx6.48$. I have recently made the following observations: ...
3
votes
3answers
530 views

A prime sequence can be partitioned into two sets of equal or consecutive sum

Denote $P[n]$ as the prime sequence $\{p_1,p_2,\cdots,p_n\}$. Conjecture: When $n=2k+1$ is odd, prime list $P[n]$ can be partitioned into two non-overlapping sublists, in which each sublist has ...
1
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0answers
45 views

closed form for a series with binomials and primes

does the series $\sum_{n=0}^\infty p^n \binom{x}{p^n}$ have a closed form ? ($p$ prime) this is a special case of $\sum_{n=0}^\infty p^n \left(\sum_{k=p^n}^{p^{n+1}-1}a_k\binom{x}{k}\right)$ with the ...
5
votes
1answer
360 views

Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that their sum of prime factors is equal, counting repeated prime factors. (The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!) So ...
3
votes
0answers
409 views

Two (strictly related) proofs by induction of inequalities

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
7
votes
3answers
524 views

Is $n = p-q$ equivalent to Goldbach's Conjecture?

One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.) Goldbach's conjecture states that every even integer greater than ...
0
votes
0answers
118 views

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

(EDIT from scratch). Let $\ \mathbf a := (a_1\ \ldots\ a_n)\ $ be an increasing non-constant arithmetic progression of odd positive numbers. The goal here is to resolve efficiently one of the two ...
5
votes
1answer
527 views

Any way to prove Prime Number Theorem using Hyperbolic Geometry? [closed]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$: $$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$ ...
4
votes
1answer
183 views

Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$ such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...
1
vote
1answer
131 views

Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that $$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...
7
votes
1answer
892 views

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 ...
12
votes
1answer
517 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 ...
2
votes
1answer
372 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. ...
4
votes
0answers
132 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
72 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 ...
0
votes
1answer
182 views

Conjectured Primality Test for Numbers of the Form $k \cdot 2^n+1$ 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
0answers
106 views

Primality Criterion for Specific Class of Numbers of the Form $k\cdot b^n-1$

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 $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...
1
vote
1answer
201 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 ...
2
votes
0answers
93 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
182 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 ...
9
votes
2answers
735 views

Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that. $$ p_n = n ...
2
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0answers
114 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.
3
votes
1answer
277 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), ...
5
votes
1answer
437 views

Recursivity of the primes

As is well-known, with the finite set of the first $n$ primes $p_1,\dots ,p_n$ one can find all primes exactly (i.e. without false positives) with the Eratosthenes Sieve in the interval ...
5
votes
1answer
397 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 ...