**5**

votes

**0**answers

252 views

### Primes for which 2 is a primitive root

I am writing a paper in which I keep referring to primes p for which 2 is a primitive root mod p and so I want to give a name for these primes. Is there a name for these primes in the literature ...

**1**

vote

**1**answer

162 views

### First Parameterized Subset of Primes that was Related to a Mathematical Result

To my knowledge, Fermat primes, i.e. primes of the form $2^{2^n}+1$ were the first to play a role in a mathematical result, namely in the characterization of constructible regular n-gons. Gauss ...

**7**

votes

**1**answer

637 views

### What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...

**6**

votes

**0**answers

321 views

### Twin Primes that are Sophie Germain Primes

Suppose $p$ is a prime such that $p + 2$ is also prime, and nothing else is known about $p$.
Is there any reason to think that this affects the probability that $p$ is also a Sophie Germain prime? ...

**3**

votes

**1**answer

151 views

### Decidability of prime gap sequences

Is the following problem undecidable?
Given a sequence of $n$ gaps $d_1,d_2,...,d_n$, does there exist a sequence of $n+1$ primes $p_1,p_2,...,p_{n+1}$ such that $p_{i+1} - p_i = d_i$ ?
If not, is ...

**27**

votes

**4**answers

3k views

### What is exceptional about the prime numbers 2 and 3?

Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this.
Permit me to define prime number the following way:
A number $n>1$ is a prime if all integers $d$ ...

**6**

votes

**1**answer

221 views

### Question about a certain class of primes

I've come across a set of primes in a problem I'm working on, and I'm wondering if there's more information available about them. I'm guessing not much, particularly since the question of infinitude ...

**1**

vote

**0**answers

216 views

### An inequality about Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...

**1**

vote

**2**answers

250 views

### Infinite play with tape, or covering the integers with prime arithmetic progressions

It is possible that a more technical version of this question has been
asked and answered in the literature. If so, then a reference is much
appreciated. I will phrase it in terms of colored tapes ...

**2**

votes

**1**answer

86 views

### Principally split primes with factors in arbitrarily small angular sectors

I wonder if the following is known:
let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where
$x, y$ ...

**3**

votes

**0**answers

129 views

### An estimate for dividing n^2 by each of the primes up to and including n, and then summing the results [closed]

I know that the asymptotic for the sum of all the primes up to n is $n^2/2\log n$. But I'm trying to find the formula (an estimate) for when $n^2$ is divided by each of the primes up to $n$, in turn ...

**3**

votes

**1**answer

479 views

### When does Merten's product theorem accurately estimate the number of coprimes in an interval?

Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:
Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For ...

**5**

votes

**2**answers

325 views

### Sum of digits of repeating end of reciprocal of prime over period is $\frac{9}{2}$

Take a prime other than 2,3 or 5 and look at the part of it that repeats in base 10. Is it true that the sum of the digits in the end divided by the period(number of repeated digits id always ...

**2**

votes

**3**answers

444 views

### Does this 'alternating' Euler product converge for all $\Re(s) > 0$?

Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...

**0**

votes

**3**answers

593 views

### Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...

**2**

votes

**2**answers

242 views

### What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$?

I know the following:
Proven: There is a prime number between $n$ and $2n$ for every integer $n>0$.
Conjectured: There is a prime number between $n^2$ and $(n+1)^2$ for every integer $n>0$.
...

**1**

vote

**2**answers

324 views

### prime zeta function when $0<s<1$ [closed]

I will not be surprised if this question seems trivial in MO but i asked it first in MathSE and i did not get an answer.
So, here it is:
I would like to know if there is a good estimate for the sum ...

**5**

votes

**1**answer

480 views

### Lines in image; are they significant to prime numbers if so how?

Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where ...

**3**

votes

**1**answer

335 views

### Giuga's Conjecture: Central or Peripheral?

An earlier MO question
highlighted
Giuga's Conjecture:
A positive integer $n>1$ is prime if and only if
$$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$
For example, for the prime $n=5$, ...

**3**

votes

**0**answers

193 views

### Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes.
For example, for $M=2$ and $N=4$ you get ...

**14**

votes

**1**answer

589 views

### Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...

**4**

votes

**1**answer

408 views

### Does this prime-gaps pattern occur infinitely often?

Let $p_n$ be the $n$-th prime.
For each integer $k \ge 0$, do there exist
an infinite number of $k+3$ consecutive primes
$(p_n, p_{n+1}, \ldots, p_{n+2+k})$
so that
(1) The gap between the 1st and ...

**-2**

votes

**1**answer

201 views

### Giuga and Carmichael numbers

If $p$ is both Giuga and Carmichael number
then its known that
$1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1} \equiv -1\pmod{p}$
is it true that
if $p$ is both Giuga and Carmichael number then
...

**0**

votes

**0**answers

166 views

### Interval containing prime numbers

Let $\varepsilon$ be an arbitrary small positive number. Can we prove that there exist an $n\in Z$ such that the interval $[2^n,(1+\varepsilon)2^n]$ contain a prime number?

**5**

votes

**0**answers

263 views

### Should I expect to see numbers this smooth?

I have a sequence $N_k$ of numbers whose growth I wish to determine, or at least
approximate nicely. When I look at the ratios of consecutive members,
I find some interesting simplifications ...

**-1**

votes

**1**answer

117 views

### Fermat pseudo prime base-3 [closed]

Good morning!
I have checked the following statement by random numbers of my choice. I am seriously looking for proof of the statement.
Statement: $m$ is said to be Fermat pseudo prime in base-3, ...

**2**

votes

**0**answers

131 views

### Analytic varieties for the primes and the twin primes

I am wondering what real and complex analysis say
about the primes and twin primes.
According to Wikipedia
analytic variety is defined locally as the set of common zeros of finitely many analytic ...

**5**

votes

**3**answers

335 views

### Weak versions of Bertrand's postulate

We are interested in the following statement:
For each $n>1$ and $x>2$ there is at least one prime $p$ satisfying $x<p<n x$.
For $n=2$ we get precisely the Bertrand's postulate which is ...

**4**

votes

**1**answer

184 views

### Consecutive Primes mod 3

Is anything known asymptotically about the binary "primes mod 3" sequence besides Dirichlet's result that 1 and 2 occur half of the time? For example, can you prove that it does not eventually cycle ...

**4**

votes

**2**answers

372 views

### Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture
Let $f(x)$ be non-constant irreducible polynomial with integer
coefficients, no fixed prime factor and positive
leading coefficient. Let $S$
be a finite set of ...

**0**

votes

**1**answer

85 views

### Joint Modular Distribution of Primes

Dirichlet's theorem shows that, for any fixed prime integer a,
"big prime numbers mod a" are uniformly distributed between
1 and a-1. If we similarly pick different prime integers
b,c,..., are these ...

**1**

vote

**0**answers

145 views

### Updated tables of maximal prime gaps? [closed]

The website http://www.trnicely.net/gaps/gaplist.html contains a long list of maximal and nonmaximal prime gaps. In this list, the largest maximal prime gap is one of length 1476:
Size: 1476
Gap ...

**7**

votes

**1**answer

339 views

### Are primes of density 0 in $a\cdot b^n+c$?

Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms ...

**1**

vote

**0**answers

88 views

### For which types of problems can one expect to use Bombieri-Vinogradov in place of GRH?

I must profess a general ignorance of problems that were once known to be true under GRH but has since became unconditional due to the Bombieri-Vinogradov theorem, but I am aware of the heuristic that ...

**4**

votes

**1**answer

174 views

### Higher dimensional generalization of the Hardy-Littlewood conjecture?

The famous Hardy-Littlewood conjecture on prime-tuples states that if $\{h_1, \cdots, h_k\} = \mathcal{H}$ is an admissible set, that is, for every prime $p$ the set $\mathcal{H}$ does not contain a ...

**15**

votes

**1**answer

851 views

### Sums of primes that are themselves prime

I'm not a math expert so this may be a trivial question; if $p_i$ is the $i$-th prime, let:
$$S(n) = \sum_{i=1}^n p_i$$
be the sum of the first $n$ primes and
$$P(n) = | \{1 \leq i \leq n \mid ...

**0**

votes

**1**answer

160 views

### Tail of singular series of Goldbach problem

Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$ ...

**13**

votes

**2**answers

1k views

### Can I express any odd number with a power of two minus a prime?

I have been running a computer program trying to see if I can represent any odd number in the form of
$$2^a - b$$
With b as a prime number. I have seen an earlier proof about Cohen and Selfridge ...

**4**

votes

**0**answers

319 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 ...

**18**

votes

**2**answers

790 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

**1**answer

426 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

**1**answer

472 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

**0**answers

283 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} ...

**16**

votes

**1**answer

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 $n=3$, ...

**7**

votes

**2**answers

334 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

**2**answers

223 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 ...

**9**

votes

**1**answer

430 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

**0**answers

95 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

**1**answer

179 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

**0**answers

255 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:
...