Tagged Questions

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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4
votes
0answers
309 views

About sign changes of Li(x)-π(x)

Given a constant $C$, which are the best known upper bounds for the number of sign changes of the function $$ f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x) $$ in the range ...
17
votes
2answers
732 views

Floors of rationals to powers: Infinite number of primes?

Let $r=a/b$ be a rational number in lowest terms, larger than $1$, and not an integer (so $b > 1$). Q. Does the sequence $$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor, ...
11
votes
1answer
394 views

Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...
8
votes
1answer
417 views

Would Elliott-Halberstam conjecture follow from GRH?

The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form ...
2
votes
0answers
271 views

A question concerning the strange arithmetic derivation

This question is related to Strange (or stupid) arithmetic derivation. The original question whether an unbounded sequence of iterates exists is still unanswered. $$n=\prod_{i=1}^{k}p_i^{\alpha_i} ...
14
votes
1answer
1k views

When is the product (1+1)(1+4)…(1+n^2) a perfect square?

This is a modification of an unanswered problem on the math StackExchange. When is the product (1+1)(1+4)…(1+n^2) a perfect square? If $(1+1)(1+4)…(1+n^2)=k^2$ then one possibility is $n=3, k=10.$ ...
7
votes
2answers
323 views

convergence in Z-hat; modulo prime power

The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16). Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$ by $a_0=b, a_{n+1}=2^{a_n}$. Prove that ...
3
votes
2answers
191 views

Is there a lower bound for the first non-trivial sequence of consecutive integers where each of the first $n$ primes is a least prime factor

Using the Chinese Remainder Theorem, it is very straight forward to find a sequence of consecutive integers starting at $x$ where each of the first $n$ prime numbers is a least prime factor for a ...
8
votes
1answer
396 views

Is $\{ p \alpha \}$ for prime $p$ dense in $[0,1]$?

Let $\alpha$ an irrational real number. It is well known that the set $\{ \{n \alpha \}|\,\, n \in \mathbb{N} \}$ is dense in$[0,1]$. ($\{x\}$ denotes the fractional part of $x$) But how to prove the ...
4
votes
0answers
83 views

On a weighted sum in a lemma for sieve methods

I'm reading James Maynard's paper "Small gaps between primes". Lemma 6.1 (p.14) in this paper confused me. This lemma was taken from Goldston-Graham-Pintz-Yildirim's paper "Small gaps between ...
2
votes
1answer
170 views

Generalizations of Chen's theorem

The two famous theorems of Jingrun Chen, both with similar proofs, state (respectively) that all sufficiently large even numbers are the sum of a prime and an element of $P_2$, and that there are ...
3
votes
0answers
211 views

Generating function for the characteristic function of prime numbers

What do we know about the generating function of $\chi(n)$ (A010051) $$ f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p $$ for $\chi(n)$ the characteristic function of the primes: ...
6
votes
1answer
245 views

Mersenne almost primes

I asked earlier whether it can be proved that infinitely many elements of $P_n$ for some positive value of $n$ (here $P_n$ refers to the set of numbers with at most $n$ prime divisors). There I ...
2
votes
0answers
210 views

Brun's Theorem for twin primes and its generalization [closed]

Brun's Theorem given in 1919 ensures that the sum of the reciprocals of the twin primes converges. Do you know a different proof of this same result? Moreover, you know if the "generalization" of it ...
0
votes
0answers
109 views

In this prime counting identity, can this limit of a sum be expressed as integrals?

Here's the identity I'm working with. $E_{0}(n,b) = 1$ $E_{k}(n,b) = \displaystyle\sum_{j=2}^{\lfloor n \rfloor}E_{k-1}(\frac{n}{j},b)-b \sum_{j=1}^{\lfloor \frac{n}{b} \rfloor}E_{k-1}(\frac{n}{j ...
6
votes
1answer
265 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{ ...
0
votes
0answers
127 views

$a_{0}$ such that $0<\lim\sup_{n\to\infty}\frac{p_{n+k}-p_{n}}{H_{k}\log^{a_{0}}(\frac{p_{n+k}+p_{n}}{2})}<\infty$

This question is somehow a follow-up from Would the following conjectures imply Cramer's conjecture? Let $g_{n,k}$ denote the quantity $p_{n+k}-p_{n}$, $s_{n,k}$ denote the quantity ...
4
votes
4answers
640 views

Prime numbers and limit ordinals

As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, ...
7
votes
1answer
488 views

Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?

For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq ...
1
vote
0answers
108 views

Which upper bound for $r_{0}(n)$ can be obtained through the Chinese Remainder theorem?

Assume Goldbach's conjecture. Then for every integer $n$ greater than one there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are prime numbers. For a given $n$, let's denote ...
19
votes
1answer
516 views

Avoiding multiples of $p$

Let $p$ be a prime number and $P=\{1,2,...,p-1\}$ In how many ways we can sum all the elements of $P$ in such a way that we will reach a multiple of $p$ only when we sum the last summand? For ...
-4
votes
1answer
141 views

$p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
21
votes
2answers
1k views

What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$

Suppose $m$ is a positive integer. A quantity of interest is $$ H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right) $$ The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...
4
votes
1answer
400 views

The number of distinct prime factors of $n\in\mathbb N$

Let $\omega(n)$ be the number of distinct prime factors of a natural number $n$. Note that $\omega(n)=0\iff n=1$, and that $\omega(24)=\omega(2^3\cdot 3^1)=2\ (\not = 4)$. (For more details, you ...
7
votes
2answers
335 views

Primes $p$ for which $pk+1$ is prime for small $k$ (or approximating Sophie Germain)

The twin prime conjecture says there are infinitely many pairs $p,p+2$ that are both prime, and although we still don't know whether it's true there's been a lot of progress recently showing that ...
11
votes
1answer
608 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}} ...
-6
votes
1answer
180 views

Is $2^{p}-1$ prime iff for $\frac{p-1}{2}$ odd positive integers $n$ below $p$, $(n+2)\vert (2^{p}+n)$? [closed]

As I was playing around with Mersenne numbers, and discovered the notion of Wagstaff prime going off Wikipedia, I started considering the sequence, for a given $odd$ prime number $p$, defined as ...
0
votes
0answers
192 views

Conjecture about distribution of primes in arithmetic progression

For my work, i need the following Conjecture: Let $N$ large number such that exist a prime number $q$ and $A>\frac{1}{2}$ such that $N^{1/2}<N^{A}\leq q-1<N.$ Then $\forall a\in\left[1,\, ...
7
votes
2answers
505 views

The equation $x^m-1=y^n+y^{n-1}+…+1$ in prime powers $x,y$

Does the equation $x^m-1=y^n+y^{n-1}+...+1$ have only finitely many solutions $(x,y,m,n)$ where $x,y$ are prime powers with $y>2$ and $m,n$ are integers with $m,n>1$? This question arose in the ...
5
votes
0answers
248 views

$n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...
0
votes
0answers
217 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 ...
5
votes
3answers
258 views

Quadratic residues and nonresidues of arbitrary patterns

Let $p_1, p_2, \dotsc, p_n$ be distinct primes, and let $\epsilon_1, \epsilon_2, \dotsc, \epsilon_n$ be an arbitrary sequence of $1$ and $-1$. There is an integer $a$ such that $\left( \frac{a}{p_1} ...
2
votes
0answers
207 views

Efficient ways to count primes satisfying Zhang's theorem

The theorem of Yitang Zhang states that there exist a finite $k \in \mathbb{N}$ such that there exist infinitely pairs of primes $(p,q)$ such that $|p - q| \leq k$. The statement that $k$ can be taken ...
6
votes
2answers
481 views

On the prime number theorem in arithmetic progression

The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$ In a similar manner ...
1
vote
0answers
121 views

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$

Determine whether if $n$ is a primitive pseudoperfect (semiperfect) number, then $\sigma(n)<2^{\sigma_0(n)}$. $\sigma_k(n)$ is the division function and $\sigma(n)=\sigma_1(n)$. A number is ...
1
vote
1answer
1k views

Calculating pisano periods for any integer

I recently stumbled across this SPOJ question: http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the pisano period of a number. After I researched my way through the web, I found ...
6
votes
0answers
229 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 ...
2
votes
4answers
662 views

Product of exponents of prime factorization

Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example, $$p(5184) = p(2^6 3^4) = 24 \;,$$ $$p(65536) = p(2^{16}) = 16 \;.$$ Define $P(n)$ as the number of iterations ...
5
votes
1answer
126 views

Short lattice vectors orthogonal to a random vector

Let $N$ be some prime number. Suppose I draw $s$ elements $g_1,..., g_s$, where each $g_i\in [N]$ is taken uniformly from some interval $I_i$ of size, say $\sqrt{N}$. Is it possible to provide a ...
49
votes
4answers
4k views

Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
11
votes
1answer
514 views

Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?

While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction $$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$ Based on what ...
6
votes
1answer
436 views

On permuted sum of squares of primes in a list

We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, ...
4
votes
0answers
216 views

Relative Densities

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count ...
11
votes
0answers
619 views

Are the twin primes the only positive double zeros of this real function?

Agno's answer was extremely helpful. For $x \in \mathbb{R}, x \ge 1$ define $$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$ By Wilson's theorem the positive integer zeros of $f(x)$ are ...
5
votes
2answers
319 views

Function with zeros plus/minus the primes

While playing with Cohen's pari script prodeulerrat found a function. For $s \in \mathbb{C}$ define $$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$ The product converges everywhere, no poles ...
4
votes
3answers
334 views

Set of primes dividing polynomials and composition

For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for ...
11
votes
2answers
1k views

An interaction between prime numbers

Let   $p_1\ p_2\ \ldots$ be the sequence of all natural prime numbers. There is a slight (just slight) but clear tendency for imitating the number of primes in an interval $(p_k;\ p_n)$   by ...
1
vote
0answers
307 views

Green-Tao style theorem for quadratic regressions (Ulam Spiral)

This is a naive question about number theory. Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a ...
5
votes
1answer
459 views

Other implications of Zhang's method

I have been reading a bit about Zhang's proof and the associated Polymath8 project. Though Tao's high level summary ...
0
votes
0answers
89 views

Sequences sharing some primitive prime divisors

Let $q=p^\alpha$ and $q'=p'^\alpha$. Moreover, define $r_i$ and $u_i$ as primitive prime divisor of $q^i-1$ and $q'^i-1$, respectively. Let $\{r_1\}=\{u_1\}$, $\{r_2\}=\{u_2\}$, $\{r_3\}=\{u_3\}$, ...