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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2
votes
1answer
85 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
0answers
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
1answer
476 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
2answers
321 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
3answers
439 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
3answers
591 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
2answers
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
2answers
317 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
1answer
477 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
1answer
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
0answers
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
1answer
581 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
1answer
406 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
1answer
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
0answers
165 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
0answers
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
1answer
116 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
0answers
128 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
3answers
333 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
1answer
181 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
2answers
364 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
1answer
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
0answers
144 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
1answer
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
0answers
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
1answer
172 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
1answer
845 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
1answer
159 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
2answers
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
0answers
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
2answers
788 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
423 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
468 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
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
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 $n=3$, ...
7
votes
2answers
333 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
221 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
1answer
428 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
93 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
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
0answers
251 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: ...
8
votes
1answer
277 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
250 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 ...
6
votes
1answer
284 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{ ...
4
votes
4answers
672 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 $\mathbb{N}$, ...
6
votes
1answer
559 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
121 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 ...
18
votes
1answer
539 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
147 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 ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
22
votes
2answers
2k 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 ...