All Questions
Tagged with nt.number-theory prime-numbers
1,808 questions
2
votes
0
answers
57
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Wieferich primes and identities for the Euler quotients of $2^n+1$ and $\frac{2^n+1}{3}$
Let $n>1$ be odd integer.
Define the Euler quotient $a(n)=\frac{2^{\varphi(n)}-1 \bmod n^2}{n}$.
Number $n$ with $a(n)=0$ is Wieferich number and if it is prime
it is Wieferich prime.
It is open ...
- 25.4k
6
votes
1
answer
2k
views
Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter
I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
- 63
1
vote
0
answers
100
views
Curious congruences modulo $4$ involving primes
We define
$$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2}
\sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$
(Searching the OEIS yielded no results.)
For $n>2$ we have the following experimental observations (...
- 17.9k
-3
votes
0
answers
70
views
Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Zeta?
This question stems from https://arxiv.org/abs/2411.19762 and the numerical observation that the best unconditional upper bound for $H_{1}:=\lim\inf_{n\to\infty}p_{n+1}-p_{n}$, namely $H_{1}^{\flat}=...
- 6,984
17
votes
2
answers
2k
views
Polynomials for natural numbers and irreducible polynomials for prime numbers?
Let $p$ be a prime and $n$ be a natural number.
Define inductively for prime numbers: $f_1(x) := 1$, $f_2(x):=x$, $f_p(x) := 1+\prod_{q\mid p-1} f_q(x)^{v_q(p-1)}$.
Is $f_p(x)$ always irreducible for ...
- 3,064
1
vote
0
answers
116
views
Can all congruences for a third-order recurrence relation hold for some composite $n$?
Let $p$ be a prime with $p \gt 3$. Consider the polynomial $f = x^3 - 3x -1$. Suppose $f$ is irreducible over $\mathbb{F}_{p}$. Let $E$ be the splitting field of $f$ over $\mathbb{F}_{p}$, and let $\...
1
vote
0
answers
77
views
Conjecture about Euler quotients related to non-Wieferich numbers $W(n)=\frac{2^n+1}{3}$
For odd natural $n$ define the Euler quotient:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
$a(n)=0$ is $n$ being Wieferich number (not necessarily prime).
For odd $n$,...
- 25.4k
3
votes
1
answer
596
views
Primes which are safe and Sophie Germain
If $p$ is a Sophie Germain prime then $2p+1$ is safe prime.
If $2p+1$ is safe prime then $p$ is Sophie Germain prime.
What is their conjectured distribution of primes $p$ which are both Sophie ...
- 13.9k
0
votes
0
answers
78
views
Factoring totient of a prime
Is it any easy to factor $p-1$ when $p$ is a prime compared to general factorization problem?
What about when $2p+1$ is also a prime?
- 13.9k
3
votes
0
answers
192
views
What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
- 20.2k
3
votes
1
answer
157
views
Three congruences for a Perrin-like sequence and pseudoprimes
Let $ V(n) $ be defined by the recurrence relation:
$$
V(n) = 3\,V(n-2) + V(n-3)
$$
with the initial conditions:
$$
V(0) = 3, \quad V(1) = 0, \quad \text{and} \quad V(2) = 6.
$$
If $ n $ is an odd ...
1
vote
1
answer
171
views
Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ squares and $b$ primes with $a+b\leq 3$?
This question is related to https://math.stackexchange.com/questions/3710032/conjecture-all-but-21-non-square-integers-are-the-sum-of-a-square-and-a-prime.
We know since Lagrange that every natural ...
- 6,984
2
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0
answers
110
views
+50
How to apply Pohlig Hellman using a very limited set of auxiliary inputs in that case?
So I was reading about Talotti, Paier, and Miculan - ECC’s Achilles’ Heel: Unveiling Weak Keys in Standardized Curves. The underlying idea is to lift the discrete logarithm problem to $\mathrm{prime}−...
- 87
1
vote
1
answer
199
views
Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical Bases"
I was reading Professor Nathanson's graduate texts in mathematics 164: "Additive Number Theory The Classical Bases" (http://www.alefenu.com/libri/nathansonbases.pdf), and I was wondering if ...
- 13
1
vote
0
answers
89
views
Test for odd prime triples in a $2p-1$ progression
Let $a(n)$ be A057326 (i.e., first member of a prime triple in a $2p-1$ progression).
Let $b(n) = B$ after $n-1$ iterations where we start with $A=n, B=1$ and for $i$ from $1$ to $n-1$ simultaneously ...
- 4,938
0
votes
0
answers
241
views
Conjecture about some recurrent primes
I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but ...
4
votes
0
answers
266
views
How dense are quotients of smooth numbers?
As usual, call a positive integer $y$-smooth if it has no prime factors greater than $y$. Write $S(x,y)$ for the set of $y$-smooth integers $\leq x$. Write $R(x,y)$ for the set of quotients $\{a/b: a,...
- 20.2k
1
vote
0
answers
127
views
Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
- 1,257
1
vote
1
answer
374
views
$\{ x/p\} $ on average
This is a vague question: Lemma 2.2 of Friedlander and Lagarias' "On the distribution in short intervals of integers having no large prime factor" says that $$\sum _{p\leq w}\left (\{ x/p\} -...
- 1,381
0
votes
0
answers
374
views
Is the Conjecture of Representing Integers as Differences of Semiprimes and Primes Extendable to Products of Distinct Primes?
Conjecture:
Let $k$ and $l$ be fixed distinct positive integers ($k≠l$). Then, for every positive integer $n$, there exist prime numbers $p_1,p_2,…,p_k∈\mathbb{P}$ and $q_1,q_2,…,q_l∈\mathbb{P}$ such ...
- 17
2
votes
1
answer
221
views
A question on signed Stirling numbers of the first kind
Let $(x)_0=1$ and $(x)_n=x(x-1)\cdots(x-n+1)$ for $n=1,2,3,\ldots$. The signed Stirling numbers of the first kind, $s(n,k)$ with $n\ge k\ge0$, are defined by
$$(x)_n=\sum_{k=0}^ns(n,k)x^k.$$
Question. ...
- 15.6k
1
vote
0
answers
170
views
Character sums over prime
Let $\chi$ be a quadratic character mod $q$. I am interested in finding the best result for how large $N$ should be such that it is guaranteed that
$$\sum_{p=1}^{N} \chi(p) \log p= o(N).$$
I am aware ...
- 499
-3
votes
1
answer
201
views
Formula for gaps between primes [closed]
The twin prime conjecture refers to:
$$
\liminf_{n\to \infty}\; p_{n+1} - p_{n} = 2.
$$
By reasoning I arrive at the following simple formula for gaps between primes:
\begin{align}
p_{...
- 13
0
votes
0
answers
169
views
On a property of prime numbers
Let $p_i$ be the $i^{\rm th}$ prime number (i.e. $p_1=2,\ p_2=3,\ p_3=5,\cdots$)
What is the function of number of combinations of $c_1,\cdots,c_n$ in terms of $n$ such that,
$$\sum_{i=1}^{n}c_ip_i\ =\...
4
votes
1
answer
240
views
Do there exist prime numbers of the form $n \cdot 2^n + 1$, when $n \in \mathbb{N}$ and $n > 1$?
Recently, I was studying prime sequences of the form $k \cdot 2^n + 1$, and I noticed that primes of the form $n \cdot 2^n + 1$ almost do not exist, except for the $n = 1$ case.
Are there other prime ...
- 125
0
votes
0
answers
102
views
Formalizing the "pseudorandomness" of primes
Many conjectures about primes seem to revolve around the idea of "primes are random". So I thought about how this "randomness" may be formally defined, and came up with the ...
- 1
2
votes
0
answers
76
views
upper and lower bounds on rowlands sequence
rowlands sequence is defined as follows
\begin{equation}
a_{n}=a_{n-1} + b_{n}
\end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural ...
3
votes
1
answer
216
views
Are there any positive integers $n$, $k$ such that $n > 2, k > 6$, and all prime factors of $n^k - 1$ are less than $n$?
I noticed that the prime factorization of $68^6 - 1$ is $3^2 \cdot 7^2 \cdot 13 \cdot 19^2 \cdot 23 \cdot 31 \cdot 67$, which makes all of its prime factors less than 68. This made me wonder the ...
- 425
1
vote
0
answers
60
views
On the parity of $(2^{\varphi(n)}-1) \bmod{n^2}$
For odd integer $n$ define the function
$$ J(n)=(2^{\varphi(n)}-1) \bmod{n^2}$$
$J(n)$ is integer in $[0,n^2-1]$ and it is divisible by $n$.
Integer $n$ is Wieferich number
iff $J(n)=0$ and if $n$ is ...
- 25.4k
5
votes
0
answers
261
views
Primes generated by cyclotomic polynomials
Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
4
votes
1
answer
409
views
On the parity of the sum $\sum_{1\le j<k\le p-1\atop p\nmid aj^2+bjk+ck^2}(aj^2+bjk+ck^2)$
QUESTION. Let $p$ be an odd prime and let $a,b,c\in\mathbb Z$. How to determine the parity of the sum
$$S_p(a,b,c)=\sum_{1\le j<k\le p-1\atop p\nmid aj^2+bjk+ck^2}(aj^2+bjk+ck^2)$$
in terms of $a,b,...
- 15.6k
7
votes
1
answer
276
views
From $\Lambda_k$ and $\Lambda$ to $\mu$ (or $\lambda$)
Let $\{a_n\}_{n=1}^\infty$, $a_n \in \mathbb{C}$, $|a_n|\leq 1$. Let $\Lambda_k = \mu \ast \log^k$; in particular, $\Lambda_1$ equals the von Mangoldt function $\Lambda$. Suppose that we have ...
- 20.2k
5
votes
1
answer
172
views
On vanishing of $p$-adic logarithms
Might be related to Wieferich primes.
Let $p$ be odd prime and define the Fermat quotient
$$F(n)=\frac{(2^{n-1} -1)}{n} \mod n=\frac{(2^{n-1} \bmod n^2 )-1}{n}$$
For integer $b$ let $L_p(b)$ be the $p$...
- 25.4k
2
votes
0
answers
108
views
Largest prime determinant of a binary matrix
Given an integer $n$, I want to prove the existence of an $n\times n$ binary matrix (with 0,1 entries), whose determinant is a prime number. What is a lower bound on the largest determinant that I ...
- 3,549
0
votes
0
answers
62
views
On base $b$ digits of $n\#$ (primorial)
Related to
normal numbers.
Let $n\#$ denote the primorial, the product of the first $n$ primes.
Q1 For all bases $b>1$, do the base $b$ digits of $n\#$ occur
with equal asymptotic frequency $\...
- 25.4k
5
votes
1
answer
811
views
A consequence of Firoozbakht's conjecture?
This is a question out of curiosity, while looking at the Firoozbakht's conjecture. It might not be research related, but as usual, I am not really sure if a question ever is research related or not, ...
- 3,064
4
votes
1
answer
403
views
Möbius square root function: existence of multiplicative and bounded function
With $\mu$ being the Möbius function, there exist infinite possibilities of square roots. For example, for each $n$ such that $\mu(n)\neq 0$ there is a choice: if $\mu(n)=-1$, we can choose to define $...
- 75
9
votes
2
answers
794
views
Why do these finite group Dedekind matrices seem to have integer spectrum when specialized to the order of group elements?
Let $p$ be a prime and let $f_p$ be the permutation on the set $\{1,2,\cdots,p-1\}$ which is given by taking inverses in $\mathbb{Z}/(p)$:
$$x \bmod(p) \mapsto \frac{1}{x} \bmod (p)$$
So for instance, ...
- 3,064
4
votes
2
answers
737
views
On finite products of $\frac{p+4}{p+2}$ with $p$ prime
Let us consider factorizations of rational numbers greater than one. For integers $a>b>0$, clearly
$$\frac ab =\prod_{n=b}^{a-1}\frac{n+1}n.$$
In view of Question 476578 and Max Alekseyev's ...
- 15.6k
1
vote
0
answers
152
views
On lacunary series connected with prime number theory
Consider the following lacunary sum with parameter $x$:
$$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$
As we can see for $x=\frac{\pi}{2}$
the sum becomes$$\sum_p\cos^2\left(\...
- 790
2
votes
1
answer
159
views
Primality of divisor sums
Let $k \geq 2$ be an integer. Put $[k] = \{1, \cdots, k\}$. Let $\mathcal{P} = \{p_1, \cdots, p_k\}$ be a set of $k$ primes. For every subset $S \subseteq [k]$ put $d_S = \prod_{j \in S} p_j$. The ...
- 26.9k
15
votes
4
answers
2k
views
Square roots and prime numbers
Definitions:
Here I present a novel conjecture using basic mathematical tools like the sum of the
divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of n ...
- 127
1
vote
0
answers
78
views
Partial sums of Möbius function and Euler characteristic of a simplicial complex for closed sets of a topology on the prime powers?
In A cell complex in number theory by Anders Björner, 2011 a number theoretic cell complex is described which has the property that the Euler characteristic is related to the Mertens function:
$$M(n) =...
- 3,064
4
votes
0
answers
145
views
Bounding an expression equivalent to Mertens function
Cross-posted from MathStackExchange, where the question is bountied but has not received any comment or answer)
Some months ago, I derived the following formula for the Merten's function $M(n)$ using ...
- 189
0
votes
0
answers
122
views
Explicit upper bounds on the number of primes up to the square of the $n^\text{th}$ prime number $p_n$
I'm looking for explicit upper bounds on the number of primes up to the square $m=p_n^2$ of the $n^\text{th}$ prime number.
Such estimates can rely on the knowledge of the exact number of primes up to ...
- 727
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
1
vote
1
answer
78
views
Minimum value of a function involving the divisor counting function
Fix any positive integer $n\in\mathbb{Z}^+,$ and consider the function $f_n : \mathbb{Z}^+\setminus\{n\}\to\mathbb{Z}^+$ given by $$f_n(t)=\sigma_0(n)+\sigma_0(t)-2\sigma_0(\gcd(n, t)),$$ where $\...
- 1,093
3
votes
0
answers
153
views
On a theorem by Iwaniec about binary quadratic polynomials representing infinitely many primes
In Theorem 1.1 (i) of http://matwbn.icm.edu.pl/ksiazki/aa/aa24/aa2451.pdf, Iwaniec showed that a certain type of quadratic polynomial $P(x,y)$ represents infinitely many primes, where $(x,y) \in \...
- 333
1
vote
0
answers
99
views
On the existence of a sequence of prime numbers satisfying a recursion relation
I am interested in the following question. I will be grateful for any reference, comment, or solution.
Let $p_1\geq 5$ be a given prime number. Does there exist an infinite sequence of prime numbers $...
- 391
8
votes
2
answers
393
views
Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime for all $n \ge 1$?
Using my computer, I found that the most of positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ is prime number after a few iterations. But exist some positive integer numbers, my ...
- 1,776