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.
2,218 questions
6
votes
2
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290
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Are primes slightly tilted towards non-squares modulo a fixed prime $p$?
Let $\mathcal P(N)$ be the set of all primes $\leq N$.
Given a fixed prime number $q$, we count the difference
$$\Delta_q(N)=\sum_{p\in\mathcal P(N)}\genfrac(){}{}q p$$
of squares and non-squares ...
- 18.5k
3
votes
1
answer
248
views
Sequences of irreducible polynomials
During some digging of mine, I once found the following recursively defined family of polynomials: $P_0=P^2+2; P_{k+1}=P_k^2-2$. Using them one can show with purely algebraic means that the 2-adic ...
5
votes
1
answer
326
views
Power-free numbers in arithmetic progressions
Since the Squarefree numbers have positive density, by Szemeredi theorem the sequence contains arbitrarily long arithmetic progressions. Note that here Green-Tao is not required. So that prompts the ...
2
votes
0
answers
99
views
Revisiting repunits and perfect powers
One of the most well known exponential diophantine equations is $(x^n-1)/(x-1)=y^p$, where $x,n,y,p$ are positive Integers with $n>2$, $x>1$, and $p$ prime. Some solutions are obtained with $x=3,...
4
votes
0
answers
123
views
Strings of noncototients
Given that the sequence of noncototients, i.e numbers not expressible as $n-\phi{(n)}$, probably has positive lower density, by Szemerédi theorem it should contain arithmetic progressions of any ...
10
votes
1
answer
483
views
Status of Mills' constant
There is a rather confusing state of affairs at Wikipedia concerning Mills' constant. The article on formula for primes mentions that It is not known whether it is irrational, but the article on Mills'...
3
votes
1
answer
497
views
On sums of a prime and a central binomial coefficient
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
- 18.1k
4
votes
2
answers
265
views
On even numbers of the form $p+p'+2^k$
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
- 18.1k
0
votes
1
answer
205
views
Whether $2n>10$ can be written as $p+p'+2^a+2^b$ with $p$ and $p'$ consecutive primes?
In a paper published in 1971, R. Crocker proved that there are infinitely many positive odd numbers not of the form $p+2^a+2^b$ with $p$ prime and $a,b\in\mathbb Z^+=\{1,2,3,\ldots\}$. The proof makes ...
- 18.1k
5
votes
1
answer
157
views
Sum of prime divisors functions
I was idly thinking today about the functions $\displaystyle f(n) = \sum_{p \mid n} p$ and $\displaystyle F(n) = \sum_{p^e \| n} ep$, respectively the "sum of prime divisors" function and ...
- 131
6
votes
0
answers
191
views
Character and exponential sums over primes under GRH
Theorem 18.13 of Montgomery and Vaughan's multiplicative number theory book says for non-principal characters (ignoring $\log $'s) $$\int _{-\delta }^{\delta }|\psi _\chi (\beta )|^2d\beta \ll \delta ...
- 1,676
4
votes
0
answers
312
views
Congruence restrictions on prime divisors of polynomial values
Consider the polynomial $f(x)= x^2+1$. Can you prove that there are infinitely many integers $x$ such that $f(x)$ has no prime divisor congruent to $1 \bmod 3$? Obviously the prime divisors are ...
8
votes
1
answer
696
views
Do all primes $>2$ hit $5$?
$2$ is a fixed point of the iteration:
$$q_{n+1}:=\min_{p|(q_n-1)^2+1} p$$
Start with $q_1>2$ prime. Does this iteration hit $5$? (min runs over primes)
6
votes
0
answers
344
views
For a prime, is there always a prime number for which it is a primitive root?
Artin’s primitive root conjecture states that for any integer $a\neq \pm1$ which is not a square,there are infinitely many primes $p$ such that $a$ is a primitive root mod $p$. By Heath-Brown's result,...
- 538
2
votes
0
answers
94
views
Can $w^2+bx^2+cy^2+dz^2$ be universal over a sparse subset of $\mathbb N$?
Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if
$$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$
then we say that $w^2+bx^2+cy^...
- 18.1k
4
votes
1
answer
452
views
Is this strengthening of the Maynard-Tao theorem on primes in admissible tuples known?
The groundbreaking work of Maynard and Tao showed the following fundamental result:
For any integer $m$, there exists a number $k(m)$ such that for any admissible set $H$ of $k$ integers (with $k$ at ...
- 43
7
votes
0
answers
302
views
Is 1001 the only palindrome which is a product of three consecutive primes?
I made a computational search for over all integers $N < 10^{27}$.
Method:
Generate a list of primes up to $10^9$
Iterate over consecutive prime triples and compute the product
Check each product ...
- 115
22
votes
2
answers
2k
views
Can every natural number be written as a product of integer powers of primes minus 1?
Do for every natural number $n$ exist (possibly negative) integers $a_p$, finitely many of them nonzero, such that
$$\log(n) = \sum_{p \text{ prime}} a_p \log(p-1)\,?$$
Equivalently:
$$n = \prod_{p \...
-5
votes
1
answer
200
views
Given a prime $p$ and by Dirichlet a prime $q = k\cdot p+1$ — minimal of this form — does the number $k = (q-1)/p$ have only prime divisors $<p$?
Given a prime $p$ and by Dirichlet a prime $q = k\cdot p+1$ - minimal of this form -,
does then the number $k = (q-1)/p$ have only prime divisors $< p$?
What does the research literature say for ...
1
vote
0
answers
61
views
Dynamics of the arithmetic–derivative family $f_k(n)=n+k(D(n)-1)$
Let $D(n)$ be the arithmetic derivative, defined by: $D(p)=1$ for primes $p$, $D(ab)=D(a)b+aD(b).$
For a fixed integer $k$, consider the dynamical system
$$f_k(n)=n+k(D(n)−1).$$
I am interested in the ...
- 1,018
-4
votes
1
answer
406
views
Linear independent prime numbers? [closed]
Given a prime $p$ Let $$\phi(p):=\sum_{p_i <p} v_{p_i}(p-1) e_i$$ where $e_i$ is the $i$-th standard basis vector, $v_p(n)$ is the valuation of $n$ for the prime $p$ and $p_i$ is the $i$-th prime ...
0
votes
0
answers
88
views
Sieve method with primes in the selected range
Is it possible to use the sieve method to solve problems like these?
Count a number of $p_1p_2\cdots p_r$, a product of $r\geqslant 1$ primes such that $p_i\in [P,2P]$ for all $i=1,2,\ldots, r$ in ...
3
votes
0
answers
174
views
Is there a group-theoretical underpinning to Giuga's conjecture?
Recall that Giuga's conjecture (1950), still widely open, asserts: let $n$ be a positive integer, if $1+\sum_{k=1}^{n-1}k^{n-1} \equiv 0\pmod{n}$ then $n$ is prime.
In light of the question and ...
- 1,347
-4
votes
1
answer
207
views
A conjecture on prime distribution (PKD Conjecture) [closed]
I would like to propose the following conjecture
The PKD Conjecture (PKD)
Let $p,d$ be positive integers with $\gcd(p,d)=1$. There exists a function
$$
f:\mathbb{N}\to\mathbb{N}, \quad f(N)<N,
$$
...
- 23
2
votes
0
answers
193
views
A provable version of alternative to Sophie Germain Primes
Sophie Germain primes $p$ satisfy $2p+1$ being prime. This is not proved.
What is the highest $\alpha$ known for provable statement for "There are infinitely many primes $p$ such that the largest ...
- 73
5
votes
1
answer
487
views
Examples of "Cell divisions" in mathematics?
I stumbled upon a mathematical structure, which I would describe as a cell division from biology, while researching prime factorization trees:
The image show a cell division:
blue = Growth of classes,...
8
votes
1
answer
393
views
On sums of finitely many distinct unit fractions of the form $\frac1{p_k+p_{k+1}}$
A well-known result on Egyptian fractions states that any positive rational number can be written as a sum of finitely many distinct unit fractions.
For each prime $p$, let $p'$ be the first prime ...
- 18.1k
2
votes
0
answers
165
views
Connected components of the graph whose vertices are the primes and in which two vertices are connected by an edge if they differ by a power of 2
Let $\Gamma$ be the graph whose vertices are the prime numbers and in which
two vertices are connected by an edge if they differ by a power of 2.
Questions:
Is it true that $\Gamma$ has exactly one ...
- 19.9k
0
votes
1
answer
205
views
Ternary Goldbach-type problems
I am looking for problems comparable to the ternary Goldbach problem, which says that every positive odd integer may be written as the sum of three primes. For instance, something of the shape
Is ...
user574181
8
votes
1
answer
350
views
Large prime divisors in cyclotomic evaluations
Let $a\ge 2$ and $n\ge 3$ be positive integers, and let
$$
\Phi_n(x) = \prod_{\substack{0 \le k < n \\ \gcd(k,n) = 1}} \left(x - e^{\frac{2\pi i k}{n}}\right)
$$
be the $n$-th cyclotomic polynomial....
- 1,116
1
vote
2
answers
182
views
Power of a primitive prime factor dividing a number of the form $x^{p}+1$
This is a lemma to solve a problem I have in mind.
Let $q$ be a primitive prime factor of $x^{p}+1$, where $x$ is a fixed positive integer and $p>11$ is a prime number. That is, a prime such that $...
3
votes
2
answers
858
views
Has an explicit value of \\(x_0\\) been computed in the Baker–Harman–Pintz theorem on prime gaps?
In Baker–Harman–Pintz (2001), “The difference between consecutive primes, II”, the authors proved that
\[
p_{n+1} - p_n \le p_n^{0.525}
\]
for all sufficiently large primes \\(p_n > x_0\\), where \\...
- 35
0
votes
0
answers
142
views
Sign function in finite field
We build a finite field $\mathbb Z/p\mathbb Z$, $p>2$. Then we introduce a sign function, which is $0$ at $0$, $+1$ at $1\dots(p-1)/2$, and $-1$ at $(p+1)/2\dots p-1$. Now we want to generalize the ...
- 109
1
vote
1
answer
105
views
A $q$-analogue of the inequality for prime gaps
Let $ p_n $ be the $ n $-th prime number, and let
$$
k = \left\lfloor \frac{p_{n+1} - p_n}{2} \right\rfloor.
$$
Consider the inequality
$$
\frac{[k+1]_q \,[k]_q}{1+q} < [p_n]_q,
$$
where the $ q $-...
- 11
3
votes
0
answers
363
views
A new generalization of Euler product formula?
I show below a formula that I've derived recently from the well-known Euler product formula, which could be considered as a generalization of it.
Let's start with a definition. For any non-empty set ...
- 153
12
votes
1
answer
1k
views
Primes whose squares add up to another square
Are there infinitely many sets of distinct primes whose squares add up to another square?
- 3,687
7
votes
2
answers
1k
views
Where is paper proving asymptotic growth of Nicolas criterion for Riemann Hypothesis?
Nicolas has shown Nicolas result that if
\begin{equation}\label{Gk}
G(k)=G_0(k)-{\rm e}^{\gamma}\ln\ln N_k>0,
\end{equation}
for all $k\ge 2$, the Riemann Hypothesis is true.
\begin{equation}
...
24
votes
1
answer
2k
views
A mysterious recurrence for primes
For any positive integer $n$, define $s(n)$ as the smallest positive integer $m$ such that the $n$ distinct numbers
$$ (p_1-1)^2,\ (p_2-1)^2,\ \ldots,\ (p_n-1)^2$$
are pairwise incongruent modulo $m$,...
- 18.1k
3
votes
0
answers
221
views
On the set $\{\sum_{k=1}^n p_k:\ n = 1,2,3,\ldots\}$
For any positive integer $n$, let $S(n)$ be the sum of the first $n$ primes. Then
$$S(1) = 2,\ S(2)=2+3=5,\ S(3)=2+3+5 =10,\ S(4) = 2+ 3+5+7 =17.$$
By the Prime Number Theorem,
$$S(n)\sim \frac{n^2}2\...
- 18.1k
2
votes
1
answer
406
views
Are there infinitely many differences of cubed primes that are perfect squares?
So the question is formulated as:
Does equation $$ p^3-q^3=x^2 $$ admits infinitely many prime solutions $p,q$ with $x\ge1$?
Some trivial analysis: it's equivalent to $(p-q)(p^2+pq+q^2)=x^2$. Note ...
- 541
1
vote
0
answers
126
views
Weighted sums of four primes
Sums of primes have been studied by number theorists for many years. Goldbach's conjecture is the most famous unsolved problem in this direction.
Here I'd like to consider weighted sums of primes. For ...
- 18.1k
6
votes
0
answers
310
views
Questions motivated by Goldbach's conjecture and the four-square theorem
Goldbach's conjecture asserts that for any integer $n>1$ we have $2n=p+q$ for some primes $p$ and $q$. A similar conjecture of Lemoine states that for any integer $n>2$ we can write $2n+1=p+2q$ ...
- 18.1k
3
votes
0
answers
134
views
Is there an algebraic notion of two primes being close?
I apologise in advance for the vagueness of the question below. I am not at all an expert in algebraic geometry, it might be that the question will come across as very naive, sorry!
I was wondering ...
- 3,118
2
votes
1
answer
158
views
Is it true that $\{p_{2^m+1}-p_{2^m}:\ m\in\mathbb Z^+\}=\{2n:\ n\in\mathbb Z^+\}$?
For $n\in\mathbb Z^+=\{1,2,3,\ldots\}$, let $p_n$ denote the $n$th prime. A well known conjecture of de Polignac states that for any $n\in\mathbb Z^+$ there are infinitely many $k\in\mathbb Z^+$ with $...
- 18.1k
1
vote
1
answer
256
views
Average of $\Lambda(n)^2$
Let $\Lambda$ be the von Mangoldt function. I am interested in understanding the average $$\sum_{n=1}^x \Lambda(n)^2.$$ By partial summation and the prime number theorem one can prove that this is $$ ...
- 3,092
1
vote
0
answers
170
views
English translation of van der Corput's 1939 proof for three-term progressions in primes
I've seen van der Corput's paper "Über Summen von Primzahlen und Primzahlquadraten" [Mathematische Annalen 116 (1939), 1–50] referenced here and there. It proves that there are infinitely ...
- 1,251
4
votes
1
answer
335
views
Divisibility property of colossally abundant numbers
The sequence of colossally abundant (CA) numbers, $a(n)$ (OEIS A004490), consists of positive integers that maximize the ratio $\frac{\sigma(m)}{m^{1+\epsilon}}$ for some $\epsilon > 0$.
A known ...
- 1,089
0
votes
0
answers
133
views
An integral related to the simple zeros conjecture
Let $$B(s)=\sum_{n\le N}b(n)n^{-s}$$ with $b(n)\ll n^{\varepsilon}.$ I want to study $$\frac{1}{2\pi i}\int_C\frac{\zeta'}{\zeta}(s)\zeta'(s)\zeta'(1-s)B(s)B(1-s)\;ds,$$ with $C$ being the contour ...
- 11
1
vote
0
answers
119
views
Heuristics for spectral norm of directed adjacency matrix connected to prime numbers?
Are there any heuristics to compute the spectral norm of the adjacency matrix of this directed graph connected to prime numbers?
Let $p$ be a prime and $n$ be a natural number.
Define inductively for ...
0
votes
0
answers
196
views
Fourth power of $\zeta(s)$ on the critical line
I am looking for a reference in the literature which gives the following form of the approximate functional equation for $|\zeta(s)|^4$.
Let $G\in C_c^{\infty}(-2,2)$ be even with $G(0)=1$, and ...
user566642