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Daniele Tampieri
Daniele Tampieri
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The groundbreaking work of Maynard and Tao showed the following fundamental result: For any integer m, there exists a number k(m) such that for any admissible set H of k integers (with k at least k(m)), there are infinitely many integers n for which the set n+H contains at least m primes. My

For any integer $m$, there exists a number $k(m)$ such that for any admissible set $H$ of $k$ integers (with $k$ at least $k(m)$), there are infinitely many integers $n$ for which the set $n+H$ contains at least $m$ primes.

My question is about a natural and stronger version. Suppose we want to force primes to appear in different parts of the set simultaneously. More More precisely: Suppose, suppose we are given numbers r$r$, and m1, m2, ..$m_1, m_2, \ldots, m_r$., mr

Question. Does there exist a constant C$C$ (depending on r$r$ and all the mj$m_j$) such that for any admissible set H$H$ of k$k$ integers (with k$k$ at least C$C$), and for any way of splitting H$H$ into r$r$ smaller groups H1, H2$H_1$, ...$H_2$, Hr$\ldots$, $H_r$, there are infinitely many integers n$n$ with the following property? For every single group Hj (where j=1 to r), the set n+Hj contains at least mj primes. In

For every single group $H_j$ (where $j=1$ to $r$), the set $n+H_j$ contains at least $m_j$ primes.

In other words, can we guarantee that we can find infinitely many n$n$ such that each part of the partition contains many primes at the same time?

The groundbreaking work of Maynard and Tao showed the following fundamental result: For any integer m, there exists a number k(m) such that for any admissible set H of k integers (with k at least k(m)), there are infinitely many integers n for which the set n+H contains at least m primes. My question is about a natural and stronger version. Suppose we want to force primes to appear in different parts of the set simultaneously. More precisely: Suppose we are given numbers r, and m1, m2, ..., mr. Does there exist a constant C (depending on r and all the mj) such that for any admissible set H of k integers (with k at least C), and for any way of splitting H into r smaller groups H1, H2, ..., Hr, there are infinitely many integers n with the following property? For every single group Hj (where j=1 to r), the set n+Hj contains at least mj primes. In other words, can we guarantee that we can find infinitely many n such that each part of the partition contains many primes at the same time?

The groundbreaking work of Maynard and Tao showed the following fundamental result:

For any integer $m$, there exists a number $k(m)$ such that for any admissible set $H$ of $k$ integers (with $k$ at least $k(m)$), there are infinitely many integers $n$ for which the set $n+H$ contains at least $m$ primes.

My question is about a natural and stronger version. Suppose we want to force primes to appear in different parts of the set simultaneously. More precisely, suppose we are given numbers $r$, and $m_1, m_2, \ldots, m_r$.

Question. Does there exist a constant $C$ (depending on $r$ and all the $m_j$) such that for any admissible set $H$ of $k$ integers (with $k$ at least $C$), and for any way of splitting $H$ into $r$ smaller groups $H_1$, $H_2$,$\ldots$, $H_r$, there are infinitely many integers $n$ with the following property?

For every single group $H_j$ (where $j=1$ to $r$), the set $n+H_j$ contains at least $m_j$ primes.

In other words, can we guarantee that we can find infinitely many $n$ such that each part of the partition contains many primes at the same time?

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mike123
mike123
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Is this strengthening of the Maynard-Tao theorem on primes in admissible tuples known?

The groundbreaking work of Maynard and Tao showed the following fundamental result: For any integer m, there exists a number k(m) such that for any admissible set H of k integers (with k at least k(m)), there are infinitely many integers n for which the set n+H contains at least m primes. My question is about a natural and stronger version. Suppose we want to force primes to appear in different parts of the set simultaneously. More precisely: Suppose we are given numbers r, and m1, m2, ..., mr. Does there exist a constant C (depending on r and all the mj) such that for any admissible set H of k integers (with k at least C), and for any way of splitting H into r smaller groups H1, H2, ..., Hr, there are infinitely many integers n with the following property? For every single group Hj (where j=1 to r), the set n+Hj contains at least mj primes. In other words, can we guarantee that we can find infinitely many n such that each part of the partition contains many primes at the same time?