All Questions
Tagged with nt.number-theory prime-numbers
1,808 questions
3
votes
0
answers
91
views
Equirepartition of sums for large multisets in subsets of finite fields
Let
$p$ be a prime number and let $\mathcal A$ be a subset of $a\leq p$ distinct
elements in $\mathbb F_p$.
We denote by $\mathcal M_k(\mathcal A)$ the set of all ${k+a-1\choose k}$
multisets ...
- 17.9k
1
vote
1
answer
286
views
GRH and the Euler product
Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
- 565
1
vote
1
answer
190
views
Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers
Let $k \in \mathbb{Z}^+$.
Is it possible to prove that, for some given
$m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$,
there are only finitely many $k$ such that the closed ...
- 1,451
0
votes
1
answer
104
views
Non-Wieferich primes with Euler quotient modulo $p$ two and alternating harmonic numbers
Let $b(n)$ denote the Euler quotient modulo $n$.
In OEIS we have A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2)
For $n>1$ we have $b(A128465(n))=2$.
...
- 25.4k
0
votes
0
answers
110
views
What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?
Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...
- 25.4k
2
votes
1
answer
283
views
Explicit bounds on number of primes of given size
How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
- 456
2
votes
0
answers
352
views
An approximation for the prime counting function
NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses.
SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, ...
- 4,985
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 $\...
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
0
votes
1
answer
713
views
Is such a generalization of the twin prime conjecture known?
Russian amateur mathematician Viktor Voevodov put forward a conjecture generalizing the conjecture about twin primes. He suggested (in a slightly different formulation) that for any finite increasing ...
- 309
1
vote
0
answers
155
views
Function involving argument of the Riemann zeta function
When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation}
f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...
- 19
1
vote
1
answer
131
views
Consecutive non-powerful integers
Pair of sequences $\ v_n\ $ and $\ U_n\ $ of integers start as in the following table:
[\begin{array}{rrrrrrrrrr}
n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \ldots \\
...
- 4,786
3
votes
1
answer
293
views
Best available bounds for $\pi(Y)-\pi(Y-X)$?
I don't know much (anything) about sieves, but as I read the section on the Selberg upper bound sieve from Greaves's Sieves in Number Theory, there is a theorem 4 which says that
If $Y\ge X \ge 2$, ...
- 107
0
votes
2
answers
288
views
Counting powerful integers. Lower bounds
Remark: The upper bounds are perhaps still more interesting; I may address them in another post.
PROBLEM: Find simple (numerically efficient) lower bounds for the number of powerful integers (...
- 4,786
1
vote
0
answers
165
views
Another Goldbach variation for odd numbers?
Lemoine's conjecture (also called Levy's conjecture according to Professor Wikipedia) states that every odd integer larger than $5$ is the sum of a prime and of twice a prime.
Dabbling in the dark art ...
- 17.9k
82
votes
3
answers
20k
views
Czelakowski's claimed proof of the Twin Prime Conjecture
It seems like the article "The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic (I)" by Janusz Czelakowski ...
- 1,083
1
vote
0
answers
98
views
Reference request for a result in additive combinatorics
Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$.
The following proposition is proved: (but I cannot find out where)
Proposition: The non-empty subset sums of $[p-1]$ are equally ...
- 3,866
2
votes
1
answer
305
views
Level spacing statistics for primes
In the preprint "Level Spacing Statistics for Primes", we have found some patterns of prime spacings, which may provide new insights on the distribution of primes:
We would like to know ...
- 149
5
votes
1
answer
234
views
What are the solutions in numbers of $xyz \mid x^n + y^n + z^n$, $x,y,z$ globally coprime
What are globally coprime integers $x,y,z\in \mathbb Z^*$ such that $xyz$ divide $x^n + y^n + z^n$?
I have no other motivation for that problem but its inherent beauty and interest.
Note that it can ...
- 687
0
votes
0
answers
68
views
Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes
This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
1
vote
1
answer
347
views
On equations with arithmetic functions [closed]
Is this good topic for research:
equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ?
If Anyone here have an advise please tell me ...
- 31
0
votes
0
answers
462
views
Relation between sieve wheel and Sundaram sieve
I made this sieve for prime numbers, which I briefly describe:
We consider $\quad p=r+modulus \cdot k \quad$ with $\quad modulus=p_1*p_2* \cdots *p_m$
and then we choose an appropriate reduced ...
- 179
0
votes
0
answers
136
views
Bounded sums involving primes
I'm trying to generalize the Theorem 2.7.1 in [1] where they prove:
$$\sum_{p \leq x} f(p) = \int_{2}^{x} \frac{f(t)}{\log{t}} dt + \epsilon(x)f(x) - \int_{2}^{x} \epsilon(t) f^{'}(t) dt $$
where $\...
- 109
6
votes
1
answer
279
views
Which $n$ have $\lvert\{2^n-2^k -1\}\cap {\mathrm{PRIMES}}\rvert=m$?
Consider numbers of the form $2^n - 2^k - 1$ with $k < n$ as considered in OEIS sequence A208083. As for A208083 I investigated how many of these numbers are prime, but turned the question around: ...
- 9,720
0
votes
2
answers
302
views
How can I convert Meissel's/Lehmer's formula for prime counting to get sum of primes
Legendre's formula can be very easily be generalised as mentioned here (visible after login) which is like this
${\pi}(v,p)={\pi}(v,p-1)-1.[{\pi}(v/p,p-1)-{\pi}(p-1,p-1)]$
${ \big\downarrow}$
$S(v,p)=...
- 103
0
votes
0
answers
113
views
If we weaken Polignac's conjecture to an existential claim, can it be proved?
Polignac's conjecture (unproved) states that, for any integer $k \geq 1$, there exist infinitely many $p$ such that $p$ and $p+2k$ are both prime. Suppose that we weaken the consequent to require only ...
- 459
0
votes
0
answers
169
views
On $\sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds$
Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...
- 11
2
votes
0
answers
93
views
Primes of the form $a+b^k$ for $k=(a \bmod 2),\ldots,n$?
Procrastinal problem: Given $n$, one can ask for integers $a,b>1$ of different parities
such that $a+b^k$ is prime for $k=(a\bmod 2),\ldots,n$.
A few examples are:
$2+4995825^k$ is prime for $k=0,\...
- 17.9k
0
votes
1
answer
1k
views
Alternative proofs of Euclid-Euler theorem
What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne ...
3
votes
0
answers
215
views
Some pending questions about $\sum_{p\leq\sqrt{n}}p=\pi(n)$
Here it was showed that $S(n)\sim \pi(n)$, where $S(n)=\sum_{p\leq\sqrt{n}}p$, $p$ refers to prime numbers, and $\pi(n)$ is the prime counting function. Here it was proved that $S(n)=\pi(n)$ for ...
- 189
0
votes
0
answers
91
views
Reducing the number of terms in Waring-Goldbach problem by allowing exponents to vary
Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing ...
- 6,984
3
votes
1
answer
228
views
What fraction of the values of a quadratic polynomial can be prime?
I have an explicit, monic quadratic polynomial $P(x)$ and an integer $m$. Can I bound the number of prime values in $P(0), P(1), \ldots, P(m)$? A reference would be appreciated, if available. An ...
- 9,114
-1
votes
1
answer
243
views
Inversion shift of a Galois radius
Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ ...
- 6,984
-10
votes
1
answer
555
views
Arithmetic billiards, prime numbers and the Goldbach conjecture
I've edited the following post on Mathematics Stack Exchange, (now closed, at that date I'm suspended) with identifier 4510963, please let me to know if you've some doubt or I can improve the post.
On ...
-10
votes
1
answer
407
views
Summatory functions for fractional parts
Notation:
$$ \{x\}\ :=\ x-\lfloor x\rfloor $$
APF-functions $\ \tau(n)\ $ for $\ 2<n\in\mathbb N,\ $ and $\ \xi(n)\ $ for $\ 3<n\in\mathbb N,\ $ are defined as follows:
$$ \tau(n)\ :=\ \sum_{k=...
- 4,786
0
votes
2
answers
132
views
Binomial congruence modulo prime [closed]
Let $a$, $b$ $(b≤a)$ be two positive integers are not twin primes and $p$ is any prime number.
Is this congruence
$$ \binom{a^p}{b^p} \equiv \binom{a}{b}^p \pmod{p} $$
valid?
- 15
4
votes
0
answers
135
views
Average of $\lambda(n+1)$ for $n$ smooth, or smooth-and-rough? What follows?
Let $\lambda$ be the Liouville function, i.e., $\lambda(p_1\dotsb p_k)=(-1)^k$ for $p_1,\dotsc,p_k$ not necessarily distinct.
There is a conjecture (due to whom?) that there are infinitely many primes ...
- 20.2k
1
vote
1
answer
153
views
Number of distinct near-squares primes dividing an odd perfect number
I'm curious about if the following question is in the literature or what work can be done about it.
Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
1
vote
2
answers
326
views
About Omega prime function
Let $ω(n)$ be the number of distinct prime factors of $n$.
Is the inequality $ω(n)\leq C\log\log(n)$ true and if so what is the value of the constant $C$ ?
- 31
0
votes
0
answers
80
views
Relevance of the deduction of similar theorems than Maier's theorem for other prime constellations
A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
3
votes
1
answer
348
views
On conjectures about the arithmetic function that counts the number of Sophie Germain primes
I've edited this post two years ago on Mathematics Stack Exchange, with identifier 3590406 and same title On conjectures about the arithmetic function that counts the number of Sophie Germain primes, ...
1
vote
0
answers
482
views
Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...
- 790
2
votes
0
answers
198
views
Generalized primality test for Mersenne and Wagstaff numbers
Inspired by the paper "Chebyshev polynomials and higher order Lucas Lehmer algorithm" by Kok Seng Chu, I think a made a generalized primality test for Mersenne and Wagstaff numbers.
Here is ...
- 119
6
votes
1
answer
267
views
Condition for $8p+1$ divides $(2^p+1)/3$?
Here is what I observed :
Let $8p+1 = 256a^2+(2b-1)^2$ with $a$ and $b$ be a positive integers, $p$ and $8p+1$ both prime numbers.
Then $8p+1$ divides $(2^p+1)/3$ only if you can write $8p+1$ as $256a^...
- 119
2
votes
0
answers
244
views
Lower bounding the number of Galois radii of an integer
Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to ...
- 6,984
-3
votes
1
answer
575
views
Digit sum of a prime number [closed]
Let 𝑝 be a positive integer and
𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that
𝑞 + 1 ≡ 0 (mod 2).
Is it that if 𝑝 is prime then 𝑞 is also prime?
e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)
- 15
1
vote
0
answers
122
views
A property related to representations of a number in prime bases
Assuming that $n>0$, let $t_b(n)$ denote the base-$b$ representation of a natural number $n$, i.e. the tuple $$(d_k, d_{k-1}, \ldots, d_1, d_0)$$ such that $$n=d_kb^k+d_{k-1}b^{k-1}+\ldots+d_1b+d_0,...
- 959
18
votes
1
answer
664
views
How hard is it to find a prime number with given primitive roots?
Assume that we randomly choose a 100-digit prime number $p$,
record which of the first 1000 prime numbers are primitive roots
modulo $p$, and then forget about $p$. —
How easy or how difficult is it ...
- 19.6k
1
vote
0
answers
107
views
Polynomial divisible by unbounded primes with exponent one
Let $f(x)$ be squarefree polynomial with integer coefficients and
degree at least $3$.
Is it true that for all sufficiently large $n$, $f(n)$ is divisible
by prime $p$ with exponent one and $p$ is ...
- 25.4k
11
votes
2
answers
1k
views
Do consecutive integers have a big prime factor?
Let us say that three consecutive positive integers $(m-1,m,m+1)$ have a big prime factor if the largest prime factor $p$ of $N=(m-1)m(m+1)$ satisfies $e^p>N$.
I ckecked that it is true for all $m&...
