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Results tagged with prime-numbers
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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.
1
vote
Prime constant graphicial representation
If I understand correctly, here is sage code to play with
(one can run in a browser in the cloud).
n=300;M=Matrix(QQ,n,n,[int(is_prime(k)) for k in xrange(1,n^2+1)]);pl=M.plot();pl.save("/tmp/primes3 …
- 25.4k
6
votes
Prime divisors of $p^n+1$
$x^n+1$ factors over $\mathbb{Z}[x]$ unless $n$ is a power of two.
For $n=15$ the factorization is $ (x + 1) \cdot (x^{2} - x + 1) \cdot (x^{4} - x^{3} + x^{2} - x + 1) \cdot (x^{8} + x^{7} - x^{5} - …
- 25.4k
3
votes
Classes of Numbers that are easy to factorize using Classical Computers?
Here are some classes of numbers that can be fully factored efficiently
with high probability.
Let $p_i$ be primes such that $2 p_i + 1$ are also primes and $q$ arbitrary
prime.
Let $n'=\prod_{i=1}^ …
- 25.4k
21
votes
Accepted
Does $2^n-n$ have infinitely often a prime divisor greater than $n$?
The answer is positive.
For natural $N$, let $n=2^N$. We are interested in large
prime factors of $2^{2^N}-2^N=2^N(2^{2^N-N}-1)$.
The second factor is of the form $2^k-1$ where $k=2^N-N$ is not
nece …
- 25.4k
4
votes
Better error bounds for partial sums of reciprocals of primes?
Under Riemann hypothesis you can get better bound.
Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$. II Lowell Schoenfeld
http://www.ams.org/journals/mcom/1976-30-134/S0025 …
- 25.4k
2
votes
0
answers
155
views
Two products over primes
For $k \in \mathbb{N}$ define
$$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$
$$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$
By the product for zeta $f(1 …
- 25.4k
1
vote
Accepted
Non primality of a pseudo-like Mersenne numbers
Yes.
For natural $k$ let $m=4+10k$.
Then $35 \times 2^m+1$ is divisible by $11$, since
$35 \cdot 2^{4+10k} \equiv -1 \pmod {11}$
and $2^m$ is periodic modulo all primes.
- 25.4k
0
votes
Infinitely many primes of the form $2^n+c$ as $n$ varies?
A heuristic approach might be to examine how many large primes are known for a given $c$.
A good source is Probable Primes Top 10000
The search form is here
Some results from the PRP Top 10000 daba …
- 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
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
2
votes
Are there open problems for primes which are known for probable primes?
There are infinitely many probable primes of the form $(n-1)^2+1$.
This generalizes to more polynomial since Fermat numbers satisfy
$F_n=(F_{n-1}-1)^2+1$.
- 25.4k
0
votes
0
answers
61
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 $\frac1b$ …
- 25.4k
2
votes
0
answers
43
views
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 probl …
- 25.4k
0
votes
1
answer
713
views
What is wrong with this counterexample to primality test assuming GRH? [closed]
From SMOOTH NUMBERS: COMPUTATIONAL NUMBER THEORY AND BEYOND Andrew Granville pp.13-14:
2j. Lenstra’s polynomial time test as to whether an integer that is conjecturally prime, is rigorously square …
- 25.4k
31
votes
Accepted
Why such an interest in studying prime gaps?
Since you ask about zeta zeros, Riemann hypothesis implies the gap is
$O(\sqrt{p_n} \log p_n)$.
Larger gap will give you nontrivial zero off the critical line,
disproving RH.
On the other hand, bou …
- 25.4k