Questions tagged [prime-numbers]
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 specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
2,020
questions
3
votes
1
answer
331
views
Prime divisors of the respectively minimal binomial coefficients
In view of Chebyshev's approach to prime numbers, I would like to ask about the regularities and peculiarities of the two sequences $\ \beta(n)\ $ and $\ \gamma(n),\ $ which I define as follows:
$\ \...
4
votes
2
answers
638
views
Conjectured relation between alternating Prime zeta series and Riemann zeta
Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad (1)$...
6
votes
1
answer
340
views
Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive ...
7
votes
2
answers
1k
views
Is there a von Koch-type theorem for the generalized Riemann hypothesis?
Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...
34
votes
2
answers
2k
views
Does iterating a certain function related to the sums of divisors eventually always result in a prime value?
Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...
6
votes
0
answers
501
views
$x^2+1$ attaining almost prime values
Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
9
votes
2
answers
1k
views
Is a Galois extension of the rationals determined by its set of completely split primes?
apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the extensions,...
8
votes
1
answer
826
views
Is it possible to sum the divergent series with prime coefficients?
It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
3
votes
0
answers
367
views
Metric on the set of subsets of the rational primes
Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version.
I was thinking how to say that two sets ...
10
votes
1
answer
385
views
Repetend digit graphs for $1/n$ in base $b$
Here is a decimal expansion of $\frac{1}{34}$:
$$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$
And here is a graphical representation of the 16-digit
"repetend," as a directed ...
3
votes
1
answer
821
views
Lower bounds on the error term of the prime number theorem
Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t.
$$f(x)\ll |\psi(x) - x|$$
where $\psi$ is the Chebyshev function.
4
votes
1
answer
968
views
Smallest prime in an arithmetic progression
Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...
11
votes
1
answer
405
views
Integers with a large prime divisor in short intervals
For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[...
8
votes
1
answer
581
views
lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?
Are there known any lower and upper bounds for
$$
\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k,
$$
where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$?
Or at least is it known ...
7
votes
2
answers
418
views
Divisor sums over values of binary forms of primes
Let $\tau$ be the divisor function, that is
$$
\tau(n)=\sharp\{d \in \mathbb{N}, d|n\}.
$$
I was wondering if anyone has ever proved an asymptotic estimate
for the sum
$$S(x):=\sum_{p,q\leq x}\tau(p^...
11
votes
1
answer
685
views
Squarefree numbers $n$ such that $432n+1$ is also squarefree
This is a second attempt (see Primes $p$ such that $432 p +1$ is prime)
Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite?
Fact: the number of such ...
0
votes
1
answer
460
views
Primes $p$ such that $432 p +1$ is prime [closed]
Is the set of prime numbers $p$ such that $432 p + 1$ is also prime infinite?
It doesn't follow from Dirichlet's theorem as far as I can tell.
6
votes
2
answers
1k
views
The shortest interval for which the prime number theorem holds [closed]
It is well known that the prime number theorem on the form
\begin{align*}
\pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)}
\end{align*}
breaks down for short enough intervals, e.g. taking $y=(\log x)^\...
0
votes
0
answers
212
views
Does $\pi(n+r)+\pi(n-r)$ decrease as $r$ increases?
Assume Goldbach's conjecture. Then for every large enough positive integer $n$ there exists a non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Such an integer $r$ will be called a ...
6
votes
1
answer
721
views
is there any heuristics suggesting that the number of Fibonacci primes below $x$ is equivalent to $\log_{\phi}\log_{\phi}x$?
The question of knowing whether there are infinitely many Fibonacci primes is an open question. As $F_p$ is prime only if $p$ is prime, one has $\pi_{FP}(x)\le \pi(\log_{\phi} x+0.5\log 5)$, but ...
21
votes
4
answers
2k
views
Prime factorization "demoted" leads to function whose fixed points are primes
Let $n$ be a natural number whose prime factorization is
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$
Define a function $g(n)$ as follows
$$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \;,$$
i.e., exponentiation is "...
4
votes
2
answers
959
views
Lower bound for a prime gap occurring infinitely often
In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound ...
16
votes
1
answer
990
views
Tight prime bounds
This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...
0
votes
0
answers
166
views
When is the earliest large prime gap also the latest large prime gap?
Suppose one finds the earliest prime gap of at least a certain size $g$, so that $p_{n+1}-p_n=g$ and $n$ is the smallest index for which the gap is as big as $g$.
Now consider the relative size of ...
1
vote
2
answers
342
views
overlap quadratic residues
Let $p$ be a prime number of form $4k+1$ and $M$ is its quadratic residue set.
Let $M_i=\{i+x|\forall x\in M\}$ $\forall 0<i<p$.
Does there exist a positive constant $\varepsilon$ such that ...
4
votes
1
answer
589
views
How frequently is 3 a cubic residue mod primes in an arithmetic progression?
Suppose $(a,3q)=1$ and $a\equiv 1\pmod 3$. Are there infinitely many primes $p\equiv a\pmod {3q}$ such that $3$ is a cubic nonresidue modulo $p$?
Or, an equivalent formulation using quadratic forms: ...
8
votes
1
answer
893
views
Is this weak asymptotic Goldbach's conjecture open?
Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.
Goldbach's conjecture: $\tau(x) = 0$
Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $
...
0
votes
2
answers
279
views
Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$
Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number.
It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...
6
votes
1
answer
1k
views
Arguments for the second Hardy–Littlewood conjecture being false?
Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that
$$\pi(x + y) - \pi(y) \leq \pi(x).$$
We can easily justify this heuristically, since
$$
\textrm{...
18
votes
1
answer
1k
views
The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?
Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...
2
votes
0
answers
614
views
Arithmetic progression and average of two prime numbers
Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1.
$$
For all terms of $A$ greater than $\ \...
5
votes
0
answers
584
views
Primality test for specific class of generalized Fermat numbers
Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $F_{p,n}= (2p)^{2^n}+1 $ where $p$ is a prime number ...
5
votes
2
answers
627
views
Primes from a Dirichlet sequence and an irrational number
From Dirichlet's theorem on arithmetic progressions, if $\text{gcd}(a,b)=1$ we know $\{ak+b\}_{k\ge 0}$ contains infinitely many primes. Let those primes be $p_1,p_2,\cdots$. Then the real
$$\alpha=0....
1
vote
0
answers
177
views
Prime Hadamard Matrices
Assume that $n$ is a sufficiently large number. Is there a Hadamaed matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, k}=1$)...
10
votes
1
answer
691
views
Primes dividing $2^a+2^b-1$
From Fermat's little theorem we know that every odd prime $p$ divides $2^a-1$ with $a=p-1$.
Is it possible to prove that there are infinitely many primes not
dividing $2^a+2^b-1$?
(With $2^a,2^...
10
votes
4
answers
2k
views
Arbitrarily long arithmetic progressions
Are there arbitrarily long arithmetic progressions in which all the
prime factors of all the terms are at most $N$, for some $N$? Assume
all the terms are positive and the sequence of terms is ...
5
votes
2
answers
263
views
Relationship of Euler product to coprimality densities for arbitrary sets of primes
Continuing the curiosity of my last couple questions: Is it the case that for every set of primes $F$, the asymptotic density of the integers coprime to all of $F$ is $\displaystyle \prod_{p \in F} (1 ...
2
votes
1
answer
342
views
Finding a suitable number
Let $n,m$ be two positive integers. By $r_n$ we denote the largest prime not exceeding $n$. If $r_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, ...
1
vote
2
answers
397
views
Consecutive primes versus prime twins
First a warm-up. Let $\ V\ $ be an arbitrary set of odd natural numbers. Let $\ S(V)\ $ be the generated multiplicative semi-group. What are the necessary and/or sufficient conditions on $\ V\ $ for ...
9
votes
5
answers
2k
views
Optical methods for number theory?
I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying
We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of ...
5
votes
1
answer
224
views
Log weight removal in general (weaker) prime number theorem
Let $a_n$ be a sequence of non-negative numbers.
Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$
Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p}{X/\...
4
votes
1
answer
210
views
Prime residua races and two views on primes
Let $\ a>1\ \ r\ \ k\ $ be arbitrary natural numbers such that $\ a\ r\ $ are relatively prime. The natural conjecture below, is it known?, is probably true in full generality:
Q1. There exists a ...
9
votes
0
answers
391
views
In which orders can the numbers of prime factors of consecutive integers be?
Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a ...
8
votes
0
answers
986
views
On the sum of consecutive primes and product of first and last
Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$
. $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$
Meaning the sum of set of ...
3
votes
3
answers
601
views
Conjecture about a sequence of natural numbers, such that, $\forall n : A_n<P_n<A_{n+1}$
Conjecture - no natural number $k$ exists such that:
$P$ is the sequence of all primes starting from the $k$th prime
$A$ is a sequence of natural numbers such that:
$\forall n : A_n<P_n<A_{n+1}...
25
votes
2
answers
2k
views
The prime numbers modulo $k$, are not periodic
Consider the sequence of prime numbers: $2,3,5,7, \cdots$. Now reduce this sequence modulo $k$ for some integer $k > 2$. Show the resulting sequence is not periodic. :
EDIT: As noted in the ...
3
votes
0
answers
746
views
New proofs of Euclid's theorem of the infinitude of primes?
Playing around with elementary inclusion-exclusion, I arrived at two simple variations of proofs of Euclid's theorem that I thought would be long known in the literature. So far I haven't been able to ...
6
votes
1
answer
368
views
Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)
Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number
of prime numbers $p \leq n$ in the residue class $r$ (mod $m$).
Further let $1 = r_1 < r_2 < \dots < r_{\varphi(m)}...
5
votes
2
answers
524
views
Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?
OEIS A226181:
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...
Primes $p$ ...
9
votes
2
answers
2k
views
Natural numbers that cannot be expressed as a difference between a square and a prime?
We wish to find the set of natural numbers that cannot be expressed as a difference between a square and a prime.
e.g.
$1 = 2^2 - 3$
$2 = 3^2 - 7$
$3 = 4^2 - 13$
and so on.
The smallest such ...