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Zhi-Wei Sun
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Let us construct a simple (undirected) graph $T$ as follows:

$\quad$ Let the set of all primes be the vertex set of $T$. For each prime $p$, take the least prime $q>p$ such that $2(p+1)-q$ is prime (such a prime q should exist in view of Goldbach's conjecture), and then set an edge connecting $p$ and $q$.

Clearly the graph $T$ contains no circle. If it is conneectedconnected then it is a tree.

QUESTION. Is the above graph $T$ a tree?

In Feb. 2013, I constructed the graph $T$ and conjectured that $T$ is indeed a tree. For example, the path connecting $2$ and $191$ is \begin{align*}2&\to 3\to 5\to 7\to 11\to 13\to 17\to 19\to 23\to 29\to 31\to 41, \\ &\to43\to 47\to 53\to 61\to 71\to 73\to 89\to 97\to 107\to 109 \\&\to 113\to 127\to 149\to 151\to 167\to 173\to 181\to 191. \end{align*}

Any ideas towards the solution of the question? Your comments are welcome!

Let us construct a simple (undirected) graph $T$ as follows:

$\quad$ Let the set of all primes be the vertex set of $T$. For each prime $p$, take the least prime $q>p$ such that $2(p+1)-q$ is prime (such a prime q should exist in view of Goldbach's conjecture), and then set an edge connecting $p$ and $q$.

Clearly the graph $T$ contains no circle. If it is conneected then it is a tree.

QUESTION. Is the above graph $T$ a tree?

In Feb. 2013, I constructed the graph $T$ and conjectured that $T$ is indeed a tree. For example, the path connecting $2$ and $191$ is \begin{align*}2&\to 3\to 5\to 7\to 11\to 13\to 17\to 19\to 23\to 29\to 31\to 41, \\ &\to43\to 47\to 53\to 61\to 71\to 73\to 89\to 97\to 107\to 109 \\&\to 113\to 127\to 149\to 151\to 167\to 173\to 181\to 191. \end{align*}

Any ideas towards the solution of the question? Your comments are welcome!

Let us construct a simple (undirected) graph $T$ as follows:

$\quad$ Let the set of all primes be the vertex set of $T$. For each prime $p$, take the least prime $q>p$ such that $2(p+1)-q$ is prime (such a prime q should exist in view of Goldbach's conjecture), and then set an edge connecting $p$ and $q$.

Clearly the graph $T$ contains no circle. If it is connected then it is a tree.

QUESTION. Is the above graph $T$ a tree?

In Feb. 2013, I constructed the graph $T$ and conjectured that $T$ is indeed a tree. For example, the path connecting $2$ and $191$ is \begin{align*}2&\to 3\to 5\to 7\to 11\to 13\to 17\to 19\to 23\to 29\to 31\to 41, \\ &\to43\to 47\to 53\to 61\to 71\to 73\to 89\to 97\to 107\to 109 \\&\to 113\to 127\to 149\to 151\to 167\to 173\to 181\to 191. \end{align*}

Any ideas towards the solution of the question? Your comments are welcome!

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Zhi-Wei Sun
  • 15.6k
  • 1
  • 20
  • 67

A tree with prime vertices

Let us construct a simple (undirected) graph $T$ as follows:

$\quad$ Let the set of all primes be the vertex set of $T$. For each prime $p$, take the least prime $q>p$ such that $2(p+1)-q$ is prime (such a prime q should exist in view of Goldbach's conjecture), and then set an edge connecting $p$ and $q$.

Clearly the graph $T$ contains no circle. If it is conneected then it is a tree.

QUESTION. Is the above graph $T$ a tree?

In Feb. 2013, I constructed the graph $T$ and conjectured that $T$ is indeed a tree. For example, the path connecting $2$ and $191$ is \begin{align*}2&\to 3\to 5\to 7\to 11\to 13\to 17\to 19\to 23\to 29\to 31\to 41, \\ &\to43\to 47\to 53\to 61\to 71\to 73\to 89\to 97\to 107\to 109 \\&\to 113\to 127\to 149\to 151\to 167\to 173\to 181\to 191. \end{align*}

Any ideas towards the solution of the question? Your comments are welcome!