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.
2,046
questions
5
votes
0
answers
58
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 $P_i(n)\in \mathbb{R}[n]$ ...
- 49.9k
0
votes
1
answer
56
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 $\...
- 1,021
-1
votes
0
answers
78
views
Possible pattern in prime numbers? [closed]
Introduction: As we know by now that there seems to be no simplistic patterns for all prime numbers. However it doesn't seem to be fair that how could a set of numbers be so arrogant in exposing ...
3
votes
0
answers
132
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
95
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 $...
- 327
8
votes
2
answers
349
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 ...
- 1,454
1
vote
0
answers
675
views
Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\pi=\dfrac{4S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
- 2,713
11
votes
0
answers
417
views
Can we rule out the possibility that $\sqrt[3]{2}$ is small modulo every prime?
Consider a prime $p$ such that the polynomial $X^3-2$ splits into linear factors over $\mathbb{F}_p$: $X^3-2 = (X-\alpha_p)(X-\beta_p)(X-\gamma_p)$. It seems reasonable to expect that (identifying $\...
- 1,894
6
votes
2
answers
353
views
About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
I already posted this question on MSE.
Using theorem $IV$ from this article, it is possible to prove that when $p$ is a prime such that $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the ...
- 365
2
votes
0
answers
152
views
The exponential sum over primes on average
In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
- 1,256
2
votes
0
answers
176
views
Integers as polynomials in infinite variables
This question is more of a request for reference or ideas than else. Forgive (or correct) if there are imprecisions or blatant mistakes.
The main idea is that the unique factorization theorem for $\...
- 29
11
votes
1
answer
462
views
Primes such that a given number has very small order
The following came up in (a previous version of) this answer.
Question. Let $a > 1$ be a positive integer, and $f \in \mathbf Z[x]$ a polynomial with positive leading term. Does there always exist ...
- 27.9k
1
vote
1
answer
160
views
Relationship between Cebotarev theorem and Lang-Trotter conjecture
The late Kevin James wrote a short survey on certain conjectures and theorems. Two of these results are retyped here.
Theorem 2 (Cebotarev). Suppose that $E/\mathbb{Q}$ is an elliptic curve and $a,m \...
- 13
3
votes
1
answer
170
views
Is there a uniform family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ such that $f_p(x)\in \mathbb{Z}[x]$ is irreducible and irreducible mod $p$?
Let $p\in\mathbb{Z}$ be a positive prime number.
Is there a "uniform" family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ of degree two such that $f_p(x)\in \mathbb{Z}[x]$ is both irreducible ...
- 237
4
votes
1
answer
246
views
Prime omega function values on a product of prime powers predecessors
Let $p_1, ... , p_n, ...$ be the prime numbers in order. Define
$$
P_n = \prod_{k=1}^n p_k^q
$$ It is known that $\omega(P_n) = n$ where $\omega(\cdot)$ is the little prime omega function. For a given,...
- 249