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Results tagged with prime-numbers
Search options questions only
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user 12481
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.
21
votes
4
answers
1k
views
Are there open problems for primes which are known for probable primes?
Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$.
Probable primes are the union of the primes and base two pseudoprimes.
This definition is much simpl …
- 25.4k
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 ex …
- 25.4k
10
votes
4
answers
1k
views
The smallest solution to $2^{2k}-1=\text{powerful}$
Integer is powerful if all the exponents in its factorization are at least $2$.
Every powerful integer can be written in the form $a^2 b^3$.
For odd $k$, define $F(k)=2^{2k}-1=(2^k-1)(2^k+1)$.
This pa …
- 25.4k
7
votes
2
answers
870
views
Unexpectedly prime rich cubic polynomial
We got a cubic polynomial which is unexpectedly prime rich.
Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and
$\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$.
Let $F(n)=\frac{\ …
- 25.4k
7
votes
2
answers
676
views
What is wrong with this deterministic algorithm efficiently generating large primes?
According to PolyMath
(Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in k …
- 25.4k
6
votes
2
answers
793
views
Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the exp …
- 25.4k
5
votes
1
answer
363
views
Prime race modulo $12$. When is the first sign change?
Define $\pi(x;q,a)$ as the number of primes less than or equal
to $x$ which are congruent to $a$ modulo $q$.
Up to $x=10^{11}$ we have $\pi(x;12,1) \le \pi(x;12,-1)$.
What is the smallest $x$ for wh …
- 25.4k
5
votes
0
answers
160
views
Reducibility of $f(x)^{2^n}+1$ and $f(x)^{2^n}+g(x)^{2^n}$
Related to generalized Fermat numbers.
Let $f(x),g(x)$ be coprime polynomials with integer coefficients.
Assume that if $f(x)$ or $g(x)$ are of the form $h(x)^k$ then $k$ is power
of two.
Q1 Is it po …
- 25.4k
5
votes
1
answer
722
views
Sum of two squares and implication of Bunyakovsky conjecture
Bunyakovsky's conjecture states that a polynomial with integer
coefficients takes infinitely many prime values at integers,
unless this is impossible for trivial reasons.
Let $a_1(x), a_2(x), a_3(x), …
- 25.4k
5
votes
2
answers
455
views
Function with zeros plus/minus the primes
While playing with Cohen's pari script prodeulerrat found a function.
For $s \in \mathbb{C}$ define
$$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$
The product converges everywhere, no poles a …
- 25.4k
5
votes
1
answer
163
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
4
votes
2
answers
460
views
Small $|2^x 3^y - 5^z 7^t|$ and generalization
Let $\{p_i\},\{q_i\}$ be disjoint sets of primes. For natural $e_i,f_i$
define $A=\prod p_i^{e_i},B=\prod q_i^{f_i}$.
Is it true that for all real $d < 1$, $|A-B| < \max(A,B)^d$
has finitely many sol …
- 25.4k
4
votes
2
answers
856
views
Can a polynomial be almost always divisible by a member of a finite set of primes?
Special case of Bunyakovsky conjecture
Let $f(x)$ be non-constant irreducible polynomial with integer
coefficients, no fixed prime factor and positive
leading coefficient. Let $S$
be a finite set of …
- 25.4k
4
votes
0
answers
187
views
Small solutions of $f(x_1,...,x_n) \equiv 0 \pmod p$
Let $f(x_1,...,x_n)$ be polynomial with integer coefficients.
Is the following possible:
For almost all primes $p$ exist integers $X_1,...,X_n$
such that:
$f(X_1,...,X_n) \ne 0$
$f(X_1,...,X_n) \equi …
- 25.4k
4
votes
2
answers
648
views
Conjectured relation between alternating Prime zeta series and Riemann zeta
Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad (1) …
- 25.4k