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2

$begingroup$

Lewis Carroll's famous game of Doublets is well known. In it you are asked to transform a given word into another by changing only one letter at a time, forming a genuine new word (not a proper name) with each letter change.

Doublets with primes is identical except that instead of playing with words you play with prime numbers, say two 3-digit primes.

Question 1. Can any 3-digit prime be transformed into any other 3-digit number following the Doublet rule?

Question 2. What is the longest distance (i.e. the largest number of links required) between two 3-digit primes?

One could ask the same questions about 4-digit primes.

graph-theory prime-numbers hamiltonian-path

share|cite|improve this question

asked 3 hours ago

Bernardo Recamán SantosBernardo Recamán Santos

410211

$endgroup$

add a comment | 

2

$begingroup$

Lewis Carroll's famous game of Doublets is well known. In it you are asked to transform a given word into another by changing only one letter at a time, forming a genuine new word (not a proper name) with each letter change.

Doublets with primes is identical except that instead of playing with words you play with prime numbers, say two 3-digit primes.

Question 1. Can any 3-digit prime be transformed into any other 3-digit number following the Doublet rule?

Question 2. What is the longest distance (i.e. the largest number of links required) between two 3-digit primes?

One could ask the same questions about 4-digit primes.

graph-theory prime-numbers hamiltonian-path

share|cite|improve this question

asked 3 hours ago

Bernardo Recamán SantosBernardo Recamán Santos

410211

$endgroup$

add a comment | 

$begingroup$

Lewis Carroll's famous game of Doublets is well known. In it you are asked to transform a given word into another by changing only one letter at a time, forming a genuine new word (not a proper name) with each letter change.

Doublets with primes is identical except that instead of playing with words you play with prime numbers, say two 3-digit primes.

Question 1. Can any 3-digit prime be transformed into any other 3-digit number following the Doublet rule?

Question 2. What is the longest distance (i.e. the largest number of links required) between two 3-digit primes?

One could ask the same questions about 4-digit primes.

graph-theory prime-numbers hamiltonian-path

share|cite|improve this question

asked 3 hours ago

Bernardo Recamán SantosBernardo Recamán Santos

410211

$endgroup$

share|cite|improve this question

asked 3 hours ago

Bernardo Recamán SantosBernardo Recamán Santos

410211

share|cite|improve this question

asked 3 hours ago

Bernardo Recamán SantosBernardo Recamán Santos

410211

asked 3 hours ago

Bernardo Recamán SantosBernardo Recamán Santos

410211

asked 3 hours ago

Bernardo Recamán SantosBernardo Recamán Santos

410211

2 Answers
2

active

oldest

votes

3

$begingroup$

I can confirm that the corresponding graph is connected. Moreover, it has a hamiltonian cycle:

share|cite|improve this answer

answered 13 mins ago

Freddy BarreraFreddy Barrera

1563

$endgroup$

add a comment | 

2

$begingroup$

For the first question the answer is yes, for the second question the answer is 6,

To solve the question i used both (Java and Wolfram), the idea is this i made a graph with nodes being the primes with 3-digits and there is a line between two nodes iff the primes representing the nodes are 1-Doublet(meaning with one digit change we can transfer one into another) and then we can state you question as graph theory question which are :

1) is the graph connected ?

2) what is the graph diameter ?

building the graph using Java and answering the questions using Wolfram we are done.

it seems that this is true for any number of digits primes, but i don't think there is a simple proof.

share|cite|improve this answer

answered 1 hour ago

AhmadAhmad

2,1071725

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Once we get to $6$ digits, the prime $294001$ has no neighbor primes, and the graph is disconnected. See weakly prime numbers.
  $endgroup$
  – Misha Lavrov
  1 hour ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MishaLavrov so for $4,5$ digits primes it works, i want to find what is the diameter!
  $endgroup$
  – Ahmad
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For 4 digits, the longest distance is 8, between 2441 and 9199 (and other pairs). For 5 digits, the longest distance is 10, between 88259 and 99721.
  $endgroup$
  – Misha Lavrov
  1 hour ago
  
  

  
  
  
  

add a comment | 

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3

$begingroup$

I can confirm that the corresponding graph is connected. Moreover, it has a hamiltonian cycle:

share|cite|improve this answer

answered 13 mins ago

Freddy BarreraFreddy Barrera

1563

$endgroup$

add a comment | 

3

$begingroup$

I can confirm that the corresponding graph is connected. Moreover, it has a hamiltonian cycle:

share|cite|improve this answer

answered 13 mins ago

Freddy BarreraFreddy Barrera

1563

$endgroup$

add a comment | 

$begingroup$

I can confirm that the corresponding graph is connected. Moreover, it has a hamiltonian cycle:

share|cite|improve this answer

answered 13 mins ago

Freddy BarreraFreddy Barrera

1563

$endgroup$

share|cite|improve this answer

answered 13 mins ago

Freddy BarreraFreddy Barrera

1563

answered 13 mins ago

Freddy BarreraFreddy Barrera

1563

answered 13 mins ago

Freddy BarreraFreddy Barrera

1563

2

$begingroup$

For the first question the answer is yes, for the second question the answer is 6,

To solve the question i used both (Java and Wolfram), the idea is this i made a graph with nodes being the primes with 3-digits and there is a line between two nodes iff the primes representing the nodes are 1-Doublet(meaning with one digit change we can transfer one into another) and then we can state you question as graph theory question which are :

1) is the graph connected ?

2) what is the graph diameter ?

building the graph using Java and answering the questions using Wolfram we are done.

it seems that this is true for any number of digits primes, but i don't think there is a simple proof.

share|cite|improve this answer

answered 1 hour ago

AhmadAhmad

2,1071725

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Once we get to $6$ digits, the prime $294001$ has no neighbor primes, and the graph is disconnected. See weakly prime numbers.
  $endgroup$
  – Misha Lavrov
  1 hour ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MishaLavrov so for $4,5$ digits primes it works, i want to find what is the diameter!
  $endgroup$
  – Ahmad
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For 4 digits, the longest distance is 8, between 2441 and 9199 (and other pairs). For 5 digits, the longest distance is 10, between 88259 and 99721.
  $endgroup$
  – Misha Lavrov
  1 hour ago
  
  

  
  
  
  

add a comment | 

2

$begingroup$

For the first question the answer is yes, for the second question the answer is 6,

To solve the question i used both (Java and Wolfram), the idea is this i made a graph with nodes being the primes with 3-digits and there is a line between two nodes iff the primes representing the nodes are 1-Doublet(meaning with one digit change we can transfer one into another) and then we can state you question as graph theory question which are :

1) is the graph connected ?

2) what is the graph diameter ?

building the graph using Java and answering the questions using Wolfram we are done.

it seems that this is true for any number of digits primes, but i don't think there is a simple proof.

share|cite|improve this answer

answered 1 hour ago

AhmadAhmad

2,1071725

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Once we get to $6$ digits, the prime $294001$ has no neighbor primes, and the graph is disconnected. See weakly prime numbers.
  $endgroup$
  – Misha Lavrov
  1 hour ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MishaLavrov so for $4,5$ digits primes it works, i want to find what is the diameter!
  $endgroup$
  – Ahmad
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For 4 digits, the longest distance is 8, between 2441 and 9199 (and other pairs). For 5 digits, the longest distance is 10, between 88259 and 99721.
  $endgroup$
  – Misha Lavrov
  1 hour ago
  
  

  
  
  
  

add a comment | 

$begingroup$

For the first question the answer is yes, for the second question the answer is 6,

To solve the question i used both (Java and Wolfram), the idea is this i made a graph with nodes being the primes with 3-digits and there is a line between two nodes iff the primes representing the nodes are 1-Doublet(meaning with one digit change we can transfer one into another) and then we can state you question as graph theory question which are :

1) is the graph connected ?

2) what is the graph diameter ?

building the graph using Java and answering the questions using Wolfram we are done.

it seems that this is true for any number of digits primes, but i don't think there is a simple proof.

share|cite|improve this answer

answered 1 hour ago

AhmadAhmad

2,1071725

$endgroup$

share|cite|improve this answer

answered 1 hour ago

AhmadAhmad

2,1071725

answered 1 hour ago

AhmadAhmad

2,1071725

answered 1 hour ago

AhmadAhmad

2,1071725