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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.
2
votes
0
answers
155
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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
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
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
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
2
votes
1
answer
218
views
Euler quotients modulo $n$
For odd integer $n$, define the Euler quotient modulo $n$ to be $a(n)$:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
$a(n)=0$ for OEIS sequence Wieferich numbers
Conje …
- 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
3
votes
1
answer
155
views
Upper bound for OEIS A076689 "Smallest k such that k*p#+1 is prime"?
OEIS A076689
Is defined as smallest integer $a(n)=k$ such that $k n\#+1$ is prime,
where $n\#$ is primorial, the product of the first $n$ primes.
Lower bound appears $1$, the primorial primes.
W …
- 25.4k
1
vote
1
answer
245
views
When is $a^{2^n}+1$ prime finitely often unconditionally?
Define generalized Fermat numbers following OEIS and mathworld.
For natural $a,n$ and $a$ even, the generalized Fermat number (GFN) is
$F_n(a)=a^{2^n}+1$.
Very large GFN primes are known (in the la …
- 25.4k
2
votes
0
answers
183
views
Claim in OEIS will give some results about Legendre's and Brocard's conjectures
Claim in OEIS will give non-trivial results about Legendre's and Brocard's conjecture. The claim is very likely to be true, but I am not sure it is currently provable.
Brocard's conjecture
states tha …
- 25.4k
3
votes
1
answer
546
views
Is there real or complex analytic function whose positive real zeros are the primes?
Related to this question
Q1 Is there real or complex analytic function $f(x)$ such
that its positive real zeros are the primes and it is
given in closed form of compositions of already named function …
- 25.4k