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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3
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1answer
256 views

Primes in short intervals with a preassigned frobenius

Edited after mistake in the first version. It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a ...
8
votes
0answers
664 views

Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...
2
votes
1answer
882 views

The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
3
votes
1answer
363 views

Heuristic for Montgomery's conjecture

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true. ...
62
votes
6answers
6k views

Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual: $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc. For $k=1,2,3,\ldots$, define $$ g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n). $$ Thus the twin ...
16
votes
1answer
4k views

Tightening Zhang's bound

Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang. The original bound was ...
4
votes
1answer
571 views

Goldbach's conjecture and Euler's idoneal numbers

Recently, I stumbled upon an interesting statement regarding Quadratic Forms. It is quite well-known and, as I will describe briefly, equivalent to Goldbach's conjecture. Let $p,q$ be odd primes and ...
19
votes
4answers
4k views

How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum

I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University. Today I found a detail in the proof ...
12
votes
2answers
1k views

Distinctive property of the primes 17 and 19?

Consider the question whether it is true that a prime number $p$ divides $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$. For the obvious heuristic reasons, for large $n$ one would ...
8
votes
3answers
509 views

Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)

What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)? In other words, I am curious about the state of the ...
2
votes
1answer
221 views

Given an even integer N, what is the minimum set of primes such that any even number x <= N can be expressed as the sum of two primes from the set?

Given an even integer N, what is the minimum set of primes such that any even number $x \leq N$ can be expressed as the sum of two primes in the set? Goldbach's conjecture said Every even integer ...
4
votes
1answer
229 views

Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...
4
votes
2answers
287 views

Are sums of the inverses of prime siblings finite?

PART I (Initial version) Let   $P$   be the set of all primes   $2\ 3\ \ldots$.   Let $$P_d\ \ :=\ \ \{\ p\in P\ :\ \exists_{q\in P}\ \ 0 < |p-q|\le d\ \}$$ and $$S_d\ :=\ ...
21
votes
0answers
1k views

Permutations of $(Z/pZ)^*$

Let $p$ be a prime integer, and let $(\mathbb Z/p\mathbb Z)^*$ be the set of non-zero elements of $\mathbb Z/p \mathbb Z$. Denote by $S((\mathbb Z/p \mathbb Z)^*)$ the group of permutations of ...
70
votes
4answers
26k 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 ...
6
votes
6answers
801 views

Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results. H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1. ...
4
votes
1answer
456 views

p such that p+1 has a large prime factor, effectively

I was reading the Boneh-Franklin IBE paper, and it seemed rather conspicuous to me that they didn't address the question of how to find primes $p$ and $q$ satisfying what they need (on page 19). ...
0
votes
1answer
411 views

A possible consequence of Dirichlet's theorem about primes in arithmetic progression

EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained. "let's consider a composite natural number $n$ greater or equal to $4$. ...
7
votes
1answer
173 views

Composing two-term sums from the same primes

The following is an old result of Erdős and Turán (American Mathematical Monthly, 1934): Given a set of $2^n + 1$ distinct positive integers, all of its two-term sums cannot be composed of the same ...
0
votes
4answers
318 views

The prime number $2$ [duplicate]

Possible Duplicate: Why is 2 so odd? I have read few books and articles, almost all of them refer that any prime $p>2$. Just wondering why it has to be $>2$?
2
votes
1answer
459 views

To express $e^{\sum \limits_{k=0}^\infty q^{2^k}}$ as product terms of $(1-q^k)^{c(k)}$

$|q|\lt1$ $A(q)=\sum \limits_{k=0}^\infty q^{2^k}$ Easily We can see that $$A(q)=q+A(q^2)\tag 1$$ Let's assume we redefine $A(q)$ as below $A(q)=-\sum \limits_{k=1}^\infty c_k \ln{(1-q^k)}$ I ...
2
votes
3answers
407 views

Is it true that $p^2+1$ is square free if $p>7$ is a Mersenne prime

For a problem in group Theory I need some information about the Mersenne primes: Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime. Is it true that $p^2+1$ is square free. i.e. if ...
5
votes
2answers
276 views

Does group of 4 equidistant successive prime exists ?

For example Group of 2 equidistant successive primes 3,5,7 distance 2 151,157,163 distance 6 Group of 3 equidistant successive primes ...
0
votes
1answer
261 views

Factorization of $p^2+1$ where $p$ is a Mersenne prime

Let $p=2^a-1$ be a Mersenne prime and so $a$ is an odd prime if $p>7$. We know that if $p=7$, then $(p^2+1)/2$ is equal to $5^2$. Can we prove that if $p>7$ , then $(p^2+1)/2$ is not equal to ...
3
votes
1answer
272 views

Least non primitive root

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally an odd prime ...
4
votes
3answers
268 views

A divergent series related to the number of divisors of of p-1

Let $d(n)$ denote the number of divisors of $n$. Is it known that the series $$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$ diverges? This would follow immediately from the Sophie Germain Conjecture. ...
1
vote
1answer
184 views

What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$

I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$. In the paper, he uses the following bounds for the second Chebyshev function ...
5
votes
2answers
316 views

special primes with p'=4p+1

How can I most quickly find a big prime, p, for which 4p+1 is also prime? For example, p=37 works. I wonder if these special primes have been researched and some characteristics are known. Are ...
4
votes
1answer
288 views

A “bit” of primes

Is there anything known/proved/conjectured about the distribution of: $$B(n) = \frac{(p_n-1)}{2} \bmod 2, \qquad p_n \mbox{ is the } n\mbox{-th prime}$$ i.e. the bit 1 of the binary representation ...
2
votes
0answers
119 views

Two products over primes

For $k \in \mathbb{N}$ define $$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$ $$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$ By the product for zeta ...
6
votes
2answers
596 views

At what point would an elementary generalization of Bertrand's Postulate be interesting?

I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$. At what point would an improvement on Nagura's result be interesting? ...
6
votes
2answers
726 views

Approximate number of primes below a given integer?

The problem of the complexity of the exact counting problem for primes is interesting. The best result we have about primes is that it is hard for TC0. But counting the number of witnesses to a TC0 ...
7
votes
0answers
272 views

Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
2
votes
1answer
320 views

At what point does Miller-Rabin become faster than trial division?

I've read in various places (and know) that Miller-Rabin is a much faster primality test than trial division for large $N$, but is much slower than trial division for small $N$. My question is: how ...
2
votes
0answers
86 views

Symmetric dominance regions surrounding a Gaussian prime

Let $z=a + b i$ be a complex number which is a Gaussian prime, on neither the $x$- nor the $y$-axis. So $a^2+b^2$ is a prime. Construct a region $D(z)$ surrounding $z$ which is the largest ...
1
vote
2answers
333 views

product 1+1/p in terms of Chebyshev's theta or psi function

I would like to know if there is any formula for $ \prod_{x<p\leq y}\left(1+\frac1p\right) $ in terms of $\theta$ or $\psi$ functions $ \theta(x)=\sum_{p\leq x}\log p $ and $ ...
2
votes
2answers
393 views

Random “pseudoprime” number generator ??

Hi How can I generate pseudoprime numbers in a given interval without... ? - going sequentially from the smallers to the largest ones (as with a Erathostenes Sieve) - nor using slow probabilistic ...
8
votes
2answers
869 views

Does this sequence exhausts the prime numbers?

We define recursively $p_1=2,p_2=3$ and $$p_{n}= \min_{(A,B)\in F_{n-1}}|A-B| $$ Where $$ \begin{split} F_n=\{(A,B) |&\gcd (A,B)=1,\quad |A-B| \not =1, \\\ &\text{both $A$ and $B$ are ...
2
votes
2answers
229 views

Catalan-type equations for prime powers

Do there exist nonzero integers $a,b,c$ for which the equation $$aX + bY = cZ$$ has infinitely many solutions with $X,Y,Z$ distinct prime powers? For example, if there are infinitely many Sophie ...
1
vote
1answer
258 views

primes of the form of 2^n+y^2

Dear All, I wonder what the numbers(primes) of the form of 2^n + x^2 (where n is even) are called? What are their properties? Any references to look at? Thank you.
5
votes
2answers
294 views

Equivalence of two well-known forms of (RH): reference-request.

This is a reference-request about a very simple statement. The Riemann hypothesis is well-known to be equivalent to $$(1)\ \ \ \pi(x) = \mathrm{Li}(x)+O(x^{1/2} \log x)$$ and to $$(2)\ \ ...
0
votes
2answers
469 views

On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients

Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ ...
2
votes
1answer
539 views

A formula combining Euler $\phi$ and $\gcd$

Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...
4
votes
0answers
318 views

AKS Algorithm Pseudoprimes

The AKS algorithm is based on the following fully deterministic primality check: Let input $n>1$ and $a \in \mathbb{N}$ such that $(a,n)=1$. Then $n$ is prime if and only if $$\tag{1}(x+a)^n ...
4
votes
3answers
598 views

Mathematical techniques to reduce the amount of storage memory

Apologies for the length question. Those acquainted with the analytics industry will know that the next big thing in the information technology world will be the Big Data revolution where huge volumes ...
27
votes
1answer
907 views

Prime Number Races in 2 Dimensions

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto \sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective? In 1999, when I was an undergraduate student, ...
22
votes
2answers
3k views

Why do primes dislike dividing the sum of all the preceding primes?

I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
6
votes
2answers
907 views

Probability that randomly chosen integers from a restricted set of natural numbers are coprime

We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is $$ P(k) = \frac{1}{\zeta(k)}. $$ I am looking at a special case of ...
0
votes
0answers
199 views

Prime numbers characterization

When, in one endeavour, I investigated prime numbers, I came up with a formula that characterizes primes, and the job was done in essentially this way: First i defined a function $sr$ (sum of ...
0
votes
0answers
186 views

Confirm/refute $f(x)$ where $f(x) = $x-th Mersenne prime ($M_p$) where $x$ is [1,2,3,4,5,6,7…]

While tinkering with numbers, I found $n=f(x)$ where $n = \sigma(\sigma(n)-n)$ and $x \neq p$ where $p$ is a Mersenne prime exponent, but I need help input in regard to improving accuracy. First off, ...