All Questions
Tagged with nt.number-theory prime-numbers
519 questions with no upvoted or accepted answers
28
votes
0
answers
716
views
Does this infinite primes snake-product converge?
This re-asks a question I posed on MSE:
Q. Does this infinite product converge?
$$
\frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
- 151k
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
19
votes
0
answers
540
views
Why $\gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}$ likes to be large?
For a prime $p$, let $F_p$ denote the greatest common divisor of the orders modulo $p$ of all prime divisors of $p-1$:
$$ F_p = \gcd \{ {\rm ord}_p(q)\colon q\mid p-1 \}; $$
thus, for instance, $F_3=...
- 23k
18
votes
0
answers
687
views
Mysterious sum equal to $\frac{7(p^2-1)}{24}$ where $p \equiv 1 \pmod{4}$
Consider a prime number $p \equiv 1 \pmod{4}$ and $n_p$ denotes the remainder of $n$ upon division by $p$. Let $A_p=\{ a \in [[0,p]] \mid {(a+1)^2}_p<{a^2}_p\}$.
I Conjecture
$$\sum_{n \in A_p } n=\...
- 1,503
17
votes
0
answers
1k
views
Colossally abundant numbers and the Riemann hypothesis
[This question followed up from a question on Math StackExchange.]
Writing Robin's inequality for the Riemann hypothesis (RH) as $$\frac{\sigma(n)}{n \ln\ln n} < e^\gamma \;,$$ we can take ...
- 229
17
votes
0
answers
891
views
An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
- 10.5k
15
votes
0
answers
365
views
Do primes of the form $4k+1$ ever lead the greatest prime factor race?
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...
- 5,470
15
votes
0
answers
487
views
Word complexity of primes mod 4
For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
- 17k
14
votes
0
answers
297
views
An 'onion-structure' for roots of a series associated to prime numbers?
The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the
sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers
defines a holomorphic function in the open disc of radius $e$.
...
- 17.9k
14
votes
0
answers
654
views
Reverse Mathematics of Euclid's theorem
Euclid's theorem that there are infinitely many prime numbers has multiple proofs, ranging from Euclid's original theorem that constructs a new prime from a finite list of such, to Euler's proof that ...
- 35.5k
14
votes
0
answers
951
views
Intersection between the sums of the first positive integers, primes and non primes
Is the following conjecture true ?
$$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap
\left\lbrace \sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}...
- 191
13
votes
0
answers
1k
views
Why am I unable to find primes of the form $(9n)!+n!+1$?
See also Math StackExchange: Is there a prime of the form $(9n)!+n!+1$?
Recently, user Peter from Math StackExchange asked for a prime of the form $(9n)!+n!+1$ (where $n$ is some natural number).
...
- 1,326
12
votes
0
answers
867
views
Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?
Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational?
The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.
- 1,805
12
votes
0
answers
704
views
Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?
Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...
- 12.6k
12
votes
0
answers
629
views
Sieve bound for prime $k$-tuples
Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by
$$
\mathfrak{S}(d_1, \ldots, ...
11
votes
0
answers
436
views
Can we rule out the possibility that $\sqrt[3]{2}$ is small modulo every prime?
Consider a prime $p$ such that the polynomial $X^3-2$ splits into linear factors over $\mathbb{F}_p$: $X^3-2 = (X-\alpha_p)(X-\beta_p)(X-\gamma_p)$. It seems reasonable to expect that (identifying $\...
- 1,914
11
votes
0
answers
458
views
effective and unconditional upper bound for the smallest quadratic residue
Let $p$ be a prime number, and let $r=r(p)$ be the smallest prime number with $(r/p)=1$. The classical result of Linnik-Vinogradov (based on Burgess) implies that $r\ll_\epsilon p^{1/4+\epsilon}$, but ...
- 1,294
11
votes
0
answers
565
views
Polynomial mapping primes to primes
Consider a non constant polynomial $P\in\mathbb{Z}[X]$ sending prime numbers to prime numbers. I encountered on the web two different proofs that $P$ is the identity polynomial, one on mathoverflow ...
- 3,425
11
votes
0
answers
524
views
Between Fermat's primes and the twin primes
Let me start with a curiosity. The integers $11,13,17,19$ are prime numbers, and $101,103,107,109$ are prime as well. One might wonder whether there is another occurrence where $10^m+1,10^m+3,10^m+7$ ...
- 52.3k
11
votes
0
answers
1k
views
Are the twin primes the only positive double zeros of this real function?
Agno's answer
was extremely helpful.
For $x \in \mathbb{R}, x \ge 1$ define
$$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$
By Wilson's theorem the positive integer zeros of $f(x)$ are ...
- 25.4k
11
votes
0
answers
2k
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, (n-r,n+...
- 6,984
10
votes
0
answers
416
views
Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
10
votes
0
answers
350
views
Are there are any attempts utilising sieve theory to attack the general $a p \pm 1$ problem?
It is currently an open question if there are infinitely many primes $p$ such that $2p + 1$ is prime (Sophie Germain primes) or that at least one of $24p \pm 1$ is prime.
Could Zhang's method, or the ...
- 533
10
votes
0
answers
205
views
Does the diophantine equation $\,\prod_{k=1}^n(p_k^{x_k}-1)=y^2\,$ have always at least a solution for $\,n\gt2\,$?
P.G.Walsh proved in this paper that the diophantine equation $\,(2^{x_1}-1)(3^{x_2}-1)=y^2\,$ has no solution in positive integers $\,x_1$, $\,x_2\,$ and $\,y$.
If we generalize the previous equation ...
- 1,008
10
votes
0
answers
633
views
Primality testing using Chebyshev polynomials
Can you provide a proof or a counterexample for the claim given below?
Inspired by an alternative definition of the Frobenius primality test which is given in this paper I have formulated the ...
- 2,661
10
votes
0
answers
269
views
On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$
I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well.
For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
- 377
10
votes
0
answers
255
views
How many partition values are expected to be prime?
Let $p(n)$ be the partition function. Let $P(N)$ count how many $1\leq n\leq N$ are such that $p(n)$ is prime.
Are there any heuristics for how $P(N)$ should behave?
A crude guess at how this ...
- 7,013
10
votes
0
answers
224
views
Product of four consecutive primes plus $1$ equals square
Some days ago, I noticed that $3\cdot 5\cdot 7\cdot 11 +1=34^2$.
I am almost sure that if we denote four consecutive primes by $p, q, r, s$ then the equation
$$p\cdot q\cdot r\cdot s+1=x^2 \quad ...
- 3,866
10
votes
0
answers
226
views
The multiplicative group generated by shifted primes
I am asking for references about the following problem.
In particular, it is still open? If not, what is the state of the art result?
Problem 1. Let $\Gamma$ be the multiplicative subgroup of $\...
user40023
10
votes
0
answers
740
views
Implications of divergence of $1/\zeta(s) $ at 1/2
$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.
This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s&...
- 2,106
10
votes
0
answers
512
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 ...
- 26k
10
votes
1
answer
449
views
On random divisor sums modulo $2^k$
Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by $...
- 265
9
votes
0
answers
201
views
primes concatenation sequence
Let us take a natural number x > 1. Then define a sequence $x_n$ as follows:
$x_0=x$;
if $x_n = p_1\cdots p_s$, where $p_1\leqslant\dots\leqslant p_s$ are prime numbers,
then $x_{n+1}$ is the ...
9
votes
0
answers
324
views
Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
- 195
9
votes
0
answers
229
views
Integers with fixed number of prime factors in arithmetic progression
Famously, there are arbitrarily long arithmetic progressions $x$, $x+y$, ..., $x+ky$ consisting of primes, by Green-Tao. I was wondering whether the following generalization is also known (by the same ...
- 2,959
9
votes
0
answers
414
views
Number of prime factors in a very short interval
Let $k \geq 3$ be a (large enough) integer, let $x \in \mathbb{R}$,
and set $I_x := [x, x + \log^k x]$.
Some believe that for $x$ large enough there exists a prime $n \in I_x$.
Equivalently, there ...
- 11.3k
9
votes
0
answers
414
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 ...
- 2,719
9
votes
0
answers
151
views
Does the given operation on pairs of primes always repeat?
Let $p$ and $q$ be two distinct primes. The set
$$A(p,q) =\{m+n : mp+nq=1 \textrm{ and } m,n \in \mathbb{Z}\}$$
is an arithmetic progression. Its step size $p-q$ is coprime to a fixed $m+n$ because
$$...
- 781
8
votes
0
answers
150
views
Can P-recursive functions assume only prime values?
A function $f\colon \{0,1,\dots\}\to \mathbb{R}$ is P-recursive if
it satisfies a recurrence $$
P_d(n)f(n+d)+P_{d-1}(n)f(n+d-1)+\cdots+P_0(n)f(n)=0,\ n\geq 0, $$
where each $P_i(n)\in \mathbb{R}[n]$ ...
- 50.8k
8
votes
0
answers
164
views
Hamiltonian paths in the prime sum graph
The following is a generalization of this old question .
Let $n\ge 2$, $[n]=\{1,\ldots,n\}$. For which distinct $a,b\in[n]$ is it possible to list $[n]$ in some order $x_1,\ldots,x_n$ such that $x_1=a$...
- 37.7k
8
votes
0
answers
289
views
A006517: Integers with $n\mid 2^n+2$
The following question was asked at Math StackExchange but, having attracted some attention, didn't get solved.
Problem 323 from the Mathematical Excalibur Vol. 14, No. 2, May-Sep. 09, linked here (...
- 550
8
votes
0
answers
341
views
k-Almost Primes in short intervals
According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...
- 1,161
8
votes
0
answers
297
views
Generating prime numbers
By a theorem of Mills, 1947, there is a real number $c$ such that for every $n$, $[c^{3^n}]$ is a prime number.
Is there a real number $d$ such that $[d^n]$ is prime, for every $n$ ?
- 1,881
8
votes
0
answers
204
views
Primes of the form $(2m+1)^2-2^{2s+1}$
The question is the following :
Question:
Does there exist infinitely many primes of the form $(2m+1)^2-2^{2s+1}$ with $m,s\geq 1$ ?
Why this could be true:
Bunyakowsky conjecture would imply ...
- 1,980
8
votes
0
answers
1k
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 ...
- 305
8
votes
0
answers
787
views
Two different ways to count Mersenne Primes
Hi there, the motivation for this question is to better understand the heuristics of Mersenne primes, and I was motivated by the recent questions (Mersenne quasi-primes) and (Primes in generalized ...
7
votes
0
answers
271
views
A question about prime numbers, totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $
This question was previously posted to MSE here.
I noticed something with the totient function $ \phi(n) $ and sum of divisors function $ \sigma(n) $ when $n > 1$.
It seems than :
$ \sigma(4n^2-1) \...
- 71
7
votes
0
answers
335
views
Residues of consecutive primes modulo a fixed integer
It is well-known that the primes are uniformly distributed in residue classes modulo any fixed integer. More precisely, for each integer $q$ and each residue $a \in \mathbb{Z}/q\mathbb{Z}$ that is ...
- 1,914
7
votes
0
answers
214
views
Does Morley's congruence characterize primes greater than $3$?
In 1895 Morley showed that $$\binom{p-1}{(p-1)/2}\equiv(-1)^{\frac{p-1}2}4^{p-1}\pmod{p^3}$$
for any prime $p>3$.
In 2009, I formulated the following conjecture concerning the converse of Morley's ...
- 15.6k
7
votes
0
answers
274
views
Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?
Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that
$$
\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0
$$
My experimental data for $n \le 6 \times 10^5 $...
- 5,470