All Questions
Tagged with nt.number-theory prime-numbers
518 questions with no upvoted or accepted answers
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A transformation game for natural numbers?
Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS ...
- 3,072
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55
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Largest interval containing family of sets with an overlap property
Here's a simplified version of a question I'm interested in.
Given $p$ and $q$ distinct prime numbers, we consider sets $A\subset \mathbb{N}\cup\{0\}, 0\in A$ of size $pq$, which are uniformly ...
- 549
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132
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Are the binary digits of the sequence of the prime numbers correlated?
Let $p_n\geq 3$ be the $n$th prime number with the binary expansion $p_n = \sum_{k=0}^{\infty} b_{nk}2^k$ ($b_{nk}\in\{0,1\}$). Let's write $q_{nk} = 1-2b_{nk}$.
Question: Is it true that for $k,l\...
- 2,605
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148
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Counting prime factors of polynomial functions
Let $\Omega(n)$ denote the number of prime factors (counted with multiplicity) of a non-zero integer $n$. For $f \in \mathbb Z[X]$ non-zero, let $$m(f) = \liminf_{n \to \infty} \Omega(f(n))$$
(1) Is $...
- 11.9k
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151
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Can every average of twin prime pairs be expressed by the sum of two smaller averages
Let's call $A$ the group of Averages of twin prime pairs (the composite between two twin primes as in OEIS A014574).
I noticed that for small numbers in $A$ they can be expressed as the sum of two ...
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234
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A set of sequences and their relations among each other
Define an order $o_p(e)$ as follows: for each even $e$, get a unique sum $p+q$ of primes $p$, $q$ (if there is one). Choose $p$ such that $p$ is the least prime in the list of primes that have been ...
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289
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Euler's totient function and primes
I'm looking for a proof of this conjecture: https://math.stackexchange.com/questions/4478597/eulers-totient-function-and-primes
$\phi$ denotes the Euler's totient function, $a$ denotes a natural ...
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179
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Getting rid of complex zeros of function with zeros the primes?
From our Note: simple real function with zeros greater than one the primes
simple real function with zeros greater than one the primes:
$j_1(x)=(\sin(\pi x))^2+(\sin(\pi \frac{\Gamma(x)+1}{x}))^2$.
...
- 25.4k
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65
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Distribution of number of prime factors of $p^k\pm1$
What is the behavior of number of prime factors of integers of form $p^k\pm1$ where $p$ is a fixed odd prime or $2$ and $k$ varies over positive integers?
- 13.9k
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155
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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
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165
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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
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98
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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
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482
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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
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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 $$n=d_kb^k+d_{k-1}b^{k-1}+\ldots+d_1b+d_0,...
- 959
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107
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Polynomial divisible by unbounded primes with exponent one
Let $f(x)$ be squarefree polynomial with integer coefficients and
degree at least $3$.
Is it true that for all sufficiently large $n$, $f(n)$ is divisible
by prime $p$ with exponent one and $p$ is ...
- 25.4k
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83
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Prime powers gap of type $(a,b)$
For $n$ a given positive integer, say $r$ is a Galois radius of $n$ of type $(a,b)$, level $l=ab$ and rank $\rho=a+b$ if $n-r=p^a$ and $n+r=q^b$ with both $p$ and $q$ prime.
Denote by $PPG_{a,b}(m)$ ...
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A definition related to pseudoprimes and the Dedekind psi function
In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
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190
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Prime numbers formed by consecutive numbers
Let $x$ be an even number. Suppose that we concatenate it with its successor to form $x (x+1)$ (not multiplication, but concatenation). For example, if we start with $x = 2$, we would get $23$. If we ...
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243
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Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?
I posted this question on SE, and was told I should repost it here.
The Goormaghtigh conjecture explores the Diophantine equation of the form
$$
\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1},
$$
where $a>c&...
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100
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Are there any connections between $a$ and $c$ where $p = a^2 + 2b^2 = c^2 + d^2$?
Let $p$ be a prime such that $p \equiv 1 \mod 8$. Then we know there exists $a,b \in \mathbb{Z}$ such that $p = a^2 + 2b^2$. But at the same time $p \equiv 1 \mod 4$, so there also exists $c,d \in \...
- 707
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241
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Liu's new sieve weight
Does Liu's sieve weight (in his arXiv paper "On the gap between primes")
$$sieve(n)=(\sum_{\substack{d_i\mid (n-h_i),i=1,\cdots,k\\ (d_1,\cdots,d_k)\in\mathcal{D}}}\lambda_{d_1,\cdots,d_k} ...
- 11
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750
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Can the prime gap record of Liu be improved further?
Let $d$ be the least positive integer such that there are infinitely many distinct prime pairs $\{p,q\}$ with $|q-p|\le d$. The twin prime conjecture is equivalent to $d=2$. In 2013 Yitang Zhang ...
- 15.6k
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66
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Is the inequality $\frac{2r_{l,0}(n)}{K_{d,0}(n)}\lesssim\log^{a+b}n$ provable for some values of $a$, $b$ and $d$?
Say $r$ is a Galois radius of level $l=ab$ and of type $(a,b)$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Let $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$ ...
- 6,984
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84
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How common are semiprimes with equally bitsized factors among semiprimes with equal bitsize?
I am curious about the following after having looked at the paper "Almost primes in almost all short intervals", theorem 3 says:
Almost all intervals $[x, x + \log^{3.51}{(x)}]$ with $x ≤ X$...
- 11
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100
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A property of $\operatorname{floor}(p/2) \bmod 2^n$ when $p$ is prime
Assuming that $i>0$, let $p(i)$ denote an $i$-th prime, i.e. $p(1)=2, p(2)=3, p(3) = 5$ etc.
Let $$f(x, y)=\operatorname{floor}(x/2) \bmod 2^y,$$ i.e. $f(123, 4)=13, f(1234567, 8)=67, f(9876543210, ...
- 959
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293
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Can a lower bound for this weakening of Goldbach's conjecture be reached?
Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a Galois radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, where $\...
- 6,984
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On a definition of sensitivity of primes in base-$2$
Given an odd integer in $\mathbb Z_{\geq0}$ of $n$ bits let $a_{n-1}a_{n-2}\dots a_1a_0$ be its binary expansion where $a_{n-1}=a_0=1$.
Call an $n$ bit prime $f(n)$-sensitive (similar to sensitivity ...
- 13.9k
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139
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Alternative Mersenne numbers
Let $\ b\in\mathbb Z,\ $ and $\ |b|>1.\ $ Call
$$ M_b(n)\ :=\ \frac{b^n-1}{b-1} $$
to be $n$-th Mersenne number mod $b$. The necessary condition for $\ M_b(n)\ $ to be a prime is that $\ n\ $ is a ...
- 4,786
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138
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The congruence $\mathrm{per}[|j-k|]_{1\le j,k\le p}\equiv-1/2\pmod p$ with $p$ an odd prime
For a matrix $[a_{j,k}]_{1\le j,k\le n}$, its permanent is given by
$$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$
Let $p$ be an odd prime. I have proved the ...
- 15.6k
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103
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$g$-gap radius of an integer
For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\...
- 6,984
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641
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Some stuff related to the twin prime conjecture
I am making three statements here, and my question is about statement 2, asking if someone can prove or disprove it. A (possibly weaker) version of statement 2 was proved as an answer to a former ...
- 3,098
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68
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Prime stretches and pulsars (alternations)
Except for prime $\ p=2,\ $ primes are divided into two disjoint and about equally frequent (Dirichlet) classes of $\ p\equiv1\mod4\ $ and $\ p\equiv-1\mod4.\ $ The natural conjecture about the ...
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Prime numbers of the form $P^x+C$
Let it be $\mathbb{P}$ the set of prime numbers. Let it be $p\in\mathbb{P}$ some prime number, and $C$ some fixed constant such that $p+C\in\mathbb{P}$. One might wonder if there exist infinitely many ...
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Smooth number pairs satisfying a congruence
Let $\mathcal P=\{p_1,\dots,p_{2t}\}$ be $2t$ primes between $2^\ell$ and $2^{\ell+1}$ and fix an exponent bound $a\in\mathbb Z_{\geq2}$.
Fix $N\in\mathbb N$ whose prime factors $p$ satisfy $p>2^{\...
- 13.9k
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Numbers whose digits, in order, display prime factors
There's a post in CodeGolf which asks for code to find numbers whose digits contain their prime factors without rearrangement. The author suggests the mathematical definition is
"Determine if ...
- 143
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An upper bound for $\,m_k=\min\,\{m\in N:\,mp_k+1\;\;is\;prime\}$
For each prime $p_k$ one can define
$$m_k=\min\,\{m\in N:\,mp_k+1\;\;is\;prime\}$$
Some computations suggest that
$$m_k=O\Big(\frac{2\sqrt k}{\log k}\Big)$$
Is this estimate confirmed by analytic ...
- 1,008
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138
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Equation $\,\ N^d\pm 1\ =\,\ p_m\cdot\ldots\cdot p_n$
Notation: $\ p_0=2,\ p_1=3,\ p_2=5, \ldots\ $ -- the increasing sequence of all primes.
(The following questions, once I've formulated them, remind me of Chebyshev). A very special case of a power $\...
- 4,786
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Generalized Thomas Ordowski conjecture at OEIS sequence A002326
OEIS is the online encyclopedia of integer sequences, Here is the link to the sequence $A002326$: https://oeis.org/A002326
For $n\geq 0$, the $n$th term in the sequence is defined as: $a(n)$ equals ...
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118
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A primality criterion for specific class of $N=4kp^n+1$
Can you provide a proof for the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ .
Let $N= 4kp^n+1 $ such that $p$ is a prime number greater ...
- 2,661
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93
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Primality test for specific class of $N=12k \cdot 5^n-1$
Can you provide a proof for the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ .
Let $N= 12k \cdot 5^{n} - 1 $ where $n\ge3$ , $12k <5^n$ ...
- 2,661
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151
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On smoothness and roughness of a number related to triangular numbers
Define $\triangle_n$ to be the $n$th triangular number.
Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$
Define $(\ell,k)$-smough numbers to be numbers that ...
- 1,826
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47
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What is the most efficient algorithm for calculating $\Phi_q(b) \operatorname{mod} N$?
From Hurwitz's theorem about irreducible factor $F_{n-1}(x)$ of degree $\varphi(n-1)$ of $x^{n-1}-1$ we can deduce the following criterion for the primality of $N=2^m \cdot p_1^{n_1} \cdot p_2^{n_2} \...
- 2,661
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133
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Primes which do not divide certain homogeneous polynomials
It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which ...
- 5,470
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77
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$t$-balanced numbers
Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture.
For $n$ a large enough composite positive integer, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...
- 6,984
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73
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How we can characterize all positive integers, multiples of 4, that cannot be expressed as $(p_1-1)(p_2-1),\;\;p_1,\,p_2$ distinct primes
I ask how we can characterize all positive integers
multiples of 4 that cannot be expressed as
$(p_1-1)(p_2-1),\;\;p_1,\,p_2$ distinct primes
The first multiples of 4 that cannot be expressed ...
- 1,008
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0
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104
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Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers
In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$
Question. Is it true that for each ...
- 15.6k
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148
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About the distribution of Fibonacci numbers that are primes
Let's consider the Fibonacci sequence, that is the sequence of naturals defined by:
$F_1=F_2=1$
$F_{n+1}=F_{n}+F_{n-1}$
It is an open problem whether the sequence contains an infinite number of ...
- 1,008
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178
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Given a prime $\,p\ne3$, is it always possible to find another prime q such that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$?
Given a prime $\,p\ne3$, is it always possible to find another prime q such
that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$ ($\,\phi\,$ is the Euler's totient function)?
Some ...
- 1,008
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315
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From an inequality for the Euler's totient function to a combination of Firoozbakht's conjecture and Nicolas' criterion for the Riemann hypothesis
In this post we ask about the veracity of an inequality deduced from a combination of Firoozbakht's conjecture (see [1] or [2]) and Nicolas' criterion for the Riemann hypothesis (see for instance [3])....
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64
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On characterizations for Mersenne primes involving the sum of divisor function
In this post we denote the sum of positive divisors function of an integer $n\geq 1$ as $$\sigma(n)=\sum_{1\leq d\mid n}d.$$
Then a prime of the form $2^p-1$ is called a Mersenne prime. These are ...