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Results tagged with prime-numbers
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user 13625
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.
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Is the upper bound on $H_{1}$ a decreasing function of the proportion of critical zeros of Z...
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}=2 …
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1
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150
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Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ sq...
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 in …
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Goldbach conjecture and the difference of two primes
This is only a partial answer. Write $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, which is well defined whenever $n>1$ if you assume Goldbach's conjecture. Then if I'm not mistaken $r_{0}(n …
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Inversion shift of a Galois radius
Only a partial answer for the special case of an integer $n$ with Galois radius $r$ of type $(a,b)$ where $\min(a,b)=1$.
Denote by $\mathbb{P}_{k}$ the set of $k$-th powers of primes and suppose $b>a$ …
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0
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0
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89
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Reducing the number of terms in Waring-Goldbach problem by allowing exponents to vary
Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing …
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241
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Inversion shift of a Galois radius
Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ is an inv …
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242
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Lower bounding the number of Galois radii of an integer
Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to prov …
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257
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Selberg's 1943 result on primes in short intervals and primality radius
This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log …
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Prime powers gap of type $(a,b)$
For $n$ a given positive integer, say $r$ is a Galois radius of $n$ of type $(a,b)$, level $l=ab$ and rank $\rho=a+b$ if $n-r=p^a$ and $n+r=q^b$ with both $p$ and $q$ prime.
Denote by $PPG_{a,b}(m)$ t …
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2
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1
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158
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Is the density of integers $n$ such that the finite sequence $(\omega(n-r)\omega(n+r))_{0\le...
Let $\omega(m)$ be the number of prime factors of $m$ regardless of multiplicity. I'm interested in the behavior of the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ for a given integer …
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0
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Is the inequality $\frac{2r_{l,0}(n)}{K_{d,0}(n)}\lesssim\log^{a+b}n$ provable for some valu...
Say $r$ is a Galois radius of level $l=ab$ and of type $(a,b)$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Let $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$ an …
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Why is there an unexpected increase in the density of certain types of Goldbach primes?
Disclaimer: this is not a real answer but rather a heuristics that is too long for a comment.
Building upon my comment, denote by $k_{0}(m):=\pi(m+r_{0}(m))-\pi(m-r_{0}(m))$ and say $m$ is $K$-central …
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Prime constellation conjectures
As a complement to Matthew Kahle's answer, see this approach to BH conjecture through Golomb's $\Lambda$-calculus by Marc Hindry and Tanguy Rivoal (in French):
https://rivoal.perso.math.cnrs.fr/articl …
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0
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1
answer
97
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$l$-th power radioprimal conjecture
I would like to know if some widely believed conjecture, be it GRH, Hardy-Littlewood conjecture, or any other would imply the following statement for some $l>1$:
$l$-th power radioprimal growth conjec …
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0
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0
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175
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Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?
Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-Litt …
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