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Smallest PRIME containing the first 11 primes as sub-strings

Smallest number containing the first 11 primes as sub-stringsThe tilted labyrinth - Can you find the fastest path in this 3D-puzzle? (Simulator now included.)The magic of the primesWhat is the smallest rectangle containing the squares of 1 through 100?Professor Halfbrain and the prime numbersA Number Game for your SoulTo Plunder Treasure IslandsFind 93 using first 4 prime numbersMake $1,dots,15$ using $3, 9, 9, 9$Smallest number containing the first 11 primes as sub-strings

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;

6

$begingroup$

In Smallest number containing the first 11 primes as sub-strings, @Alconja successfully found the smallest number which contains the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31) as concatenated sub-strings. This inspired me to propose the following followup:

What is the smallest prime which contains each of the first eleven primes as a sub-string?

Obviously the answer is at least

113,171,923,295,

but that's not prime. How much further do we need to go?

_{Disclaimer: I don't know the answer myself. I'm hoping it won't need a computer to find ...}

mathematics optimization number-theory

share|improve this question

asked 8 hours ago

Rand al'ThorRand al'Thor

76.2k1515 gold badges250250 silver badges501501 bronze badges

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  This seems like it will be very difficult to do without a computer
  $endgroup$
  – Cruncher
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  How do you find any prime greater than the spoiler value without using a computer?
  $endgroup$
  – Weather Vane
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @WeatherVane I compared my guesses to an already calculated List of primes. It was probably done with a computer, but not by me.
  $endgroup$
  – Darrel Hoffman
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DarrelHoffman I know (and have used) similar lists but didn't know they went that far.
  $endgroup$
  – Weather Vane
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  We should calculate the first N of these and submit it to the OEIS. I'll start: 2, 23, 253, 2357, 211573, 511327, 1135217... (trial and error on these, might not be all correct)
  $endgroup$
  – Darrel Hoffman
  7 hours ago
  

  
  
  
  

 | 
show 2 more comments

6

$begingroup$

In Smallest number containing the first 11 primes as sub-strings, @Alconja successfully found the smallest number which contains the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31) as concatenated sub-strings. This inspired me to propose the following followup:

What is the smallest prime which contains each of the first eleven primes as a sub-string?

Obviously the answer is at least

113,171,923,295,

but that's not prime. How much further do we need to go?

_{Disclaimer: I don't know the answer myself. I'm hoping it won't need a computer to find ...}

mathematics optimization number-theory

share|improve this question

asked 8 hours ago

Rand al'ThorRand al'Thor

76.2k1515 gold badges250250 silver badges501501 bronze badges

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  This seems like it will be very difficult to do without a computer
  $endgroup$
  – Cruncher
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  How do you find any prime greater than the spoiler value without using a computer?
  $endgroup$
  – Weather Vane
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @WeatherVane I compared my guesses to an already calculated List of primes. It was probably done with a computer, but not by me.
  $endgroup$
  – Darrel Hoffman
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DarrelHoffman I know (and have used) similar lists but didn't know they went that far.
  $endgroup$
  – Weather Vane
  8 hours ago
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  We should calculate the first N of these and submit it to the OEIS. I'll start: 2, 23, 253, 2357, 211573, 511327, 1135217... (trial and error on these, might not be all correct)
  $endgroup$
  – Darrel Hoffman
  7 hours ago
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

In Smallest number containing the first 11 primes as sub-strings, @Alconja successfully found the smallest number which contains the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31) as concatenated sub-strings. This inspired me to propose the following followup:

What is the smallest prime which contains each of the first eleven primes as a sub-string?

Obviously the answer is at least

113,171,923,295,

but that's not prime. How much further do we need to go?

_{Disclaimer: I don't know the answer myself. I'm hoping it won't need a computer to find ...}

mathematics optimization number-theory

share|improve this question

asked 8 hours ago

Rand al'ThorRand al'Thor

76.2k1515 gold badges250250 silver badges501501 bronze badges

$endgroup$

share|improve this question

asked 8 hours ago

Rand al'ThorRand al'Thor

76.2k1515 gold badges250250 silver badges501501 bronze badges

share|improve this question

asked 8 hours ago

Rand al'ThorRand al'Thor

76.2k1515 gold badges250250 silver badges501501 bronze badges

asked 8 hours ago

Rand al'ThorRand al'Thor

76.2k1515 gold badges250250 silver badges501501 bronze badges

asked 8 hours ago

Rand al'ThorRand al'Thor

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3 Answers
3

active

oldest

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9

$begingroup$

So I can't yet prove this is the smallest, but it's at least an upper bound:

113,175,192,329

Reasoning:

Obviously, we have to get that 5 away from the last digit or else it's a multiple of 5. But we can't break up the 29, 23, or 19 or we lose those primes. So I tried moving the 5 back a few digits. ‭113,171,923,529‬ is divisible by 7. 113,171,952,329 is divisible by 337. But 113,175,192,329 is prime. Might be able to improve on that with some other permutations...

share|improve this answer

answered 8 hours ago

Darrel HoffmanDarrel Hoffman

2,6991111 silver badges2626 bronze badges

$endgroup$

add a comment
 | 

8

$begingroup$

Shuffling the sequence of 5 and the non- overlapping 19, 23, and 29 by trial and error produces:

113,172,923,519

share|improve this answer

answered 7 hours ago

collapsarcollapsar

39311 silver badge66 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I figured it could go smaller, just didn't have time to play around with it anymore...
  $endgroup$
  – Darrel Hoffman
  7 hours ago
  

  
  
  
  

add a comment
 | 

2

$begingroup$

The answer is

113,171,952,923

I wrote a Java program to find it:

The program uses brute force by starting with the lower bound obtained in the previous question (113,171,923,295) and finding the next prime that contains the required primes as substrings. It turns out that we only need to check 29628 possibilities, which is not many. Here is the program: https://pastebin.com/XQL6VGnc

share|improve this answer

answered 1 hour ago

Dmitry KamenetskyDmitry Kamenetsky

96022 silver badges1818 bronze badges

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add a comment
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9

$begingroup$

So I can't yet prove this is the smallest, but it's at least an upper bound:

113,175,192,329

Reasoning:

Obviously, we have to get that 5 away from the last digit or else it's a multiple of 5. But we can't break up the 29, 23, or 19 or we lose those primes. So I tried moving the 5 back a few digits. ‭113,171,923,529‬ is divisible by 7. 113,171,952,329 is divisible by 337. But 113,175,192,329 is prime. Might be able to improve on that with some other permutations...

share|improve this answer

answered 8 hours ago

Darrel HoffmanDarrel Hoffman

2,6991111 silver badges2626 bronze badges

$endgroup$

add a comment
 | 

9

$begingroup$

So I can't yet prove this is the smallest, but it's at least an upper bound:

113,175,192,329

Reasoning:

Obviously, we have to get that 5 away from the last digit or else it's a multiple of 5. But we can't break up the 29, 23, or 19 or we lose those primes. So I tried moving the 5 back a few digits. ‭113,171,923,529‬ is divisible by 7. 113,171,952,329 is divisible by 337. But 113,175,192,329 is prime. Might be able to improve on that with some other permutations...

share|improve this answer

answered 8 hours ago

Darrel HoffmanDarrel Hoffman

2,6991111 silver badges2626 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

So I can't yet prove this is the smallest, but it's at least an upper bound:

113,175,192,329

Reasoning:

Obviously, we have to get that 5 away from the last digit or else it's a multiple of 5. But we can't break up the 29, 23, or 19 or we lose those primes. So I tried moving the 5 back a few digits. ‭113,171,923,529‬ is divisible by 7. 113,171,952,329 is divisible by 337. But 113,175,192,329 is prime. Might be able to improve on that with some other permutations...

share|improve this answer

answered 8 hours ago

Darrel HoffmanDarrel Hoffman

2,6991111 silver badges2626 bronze badges

$endgroup$

share|improve this answer

answered 8 hours ago

Darrel HoffmanDarrel Hoffman

2,6991111 silver badges2626 bronze badges

answered 8 hours ago

Darrel HoffmanDarrel Hoffman

2,6991111 silver badges2626 bronze badges

answered 8 hours ago

Darrel HoffmanDarrel Hoffman

2,6991111 silver badges2626 bronze badges

8

$begingroup$

Shuffling the sequence of 5 and the non- overlapping 19, 23, and 29 by trial and error produces:

113,172,923,519

share|improve this answer

answered 7 hours ago

collapsarcollapsar

39311 silver badge66 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I figured it could go smaller, just didn't have time to play around with it anymore...
  $endgroup$
  – Darrel Hoffman
  7 hours ago
  

  
  
  
  

add a comment
 | 

8

$begingroup$

Shuffling the sequence of 5 and the non- overlapping 19, 23, and 29 by trial and error produces:

113,172,923,519

share|improve this answer

answered 7 hours ago

collapsarcollapsar

39311 silver badge66 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I figured it could go smaller, just didn't have time to play around with it anymore...
  $endgroup$
  – Darrel Hoffman
  7 hours ago
  

  
  
  
  

add a comment
 | 

$begingroup$

Shuffling the sequence of 5 and the non- overlapping 19, 23, and 29 by trial and error produces:

113,172,923,519

share|improve this answer

answered 7 hours ago

collapsarcollapsar

39311 silver badge66 bronze badges

$endgroup$

share|improve this answer

answered 7 hours ago

collapsarcollapsar

39311 silver badge66 bronze badges

answered 7 hours ago

collapsarcollapsar

39311 silver badge66 bronze badges

answered 7 hours ago

collapsarcollapsar

39311 silver badge66 bronze badges

2

$begingroup$

The answer is

113,171,952,923

I wrote a Java program to find it:

The program uses brute force by starting with the lower bound obtained in the previous question (113,171,923,295) and finding the next prime that contains the required primes as substrings. It turns out that we only need to check 29628 possibilities, which is not many. Here is the program: https://pastebin.com/XQL6VGnc

share|improve this answer

answered 1 hour ago

Dmitry KamenetskyDmitry Kamenetsky

96022 silver badges1818 bronze badges

$endgroup$

add a comment
 | 

2

$begingroup$

The answer is

113,171,952,923

I wrote a Java program to find it:

The program uses brute force by starting with the lower bound obtained in the previous question (113,171,923,295) and finding the next prime that contains the required primes as substrings. It turns out that we only need to check 29628 possibilities, which is not many. Here is the program: https://pastebin.com/XQL6VGnc

share|improve this answer

answered 1 hour ago

Dmitry KamenetskyDmitry Kamenetsky

96022 silver badges1818 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

The answer is

113,171,952,923

I wrote a Java program to find it:

The program uses brute force by starting with the lower bound obtained in the previous question (113,171,923,295) and finding the next prime that contains the required primes as substrings. It turns out that we only need to check 29628 possibilities, which is not many. Here is the program: https://pastebin.com/XQL6VGnc

share|improve this answer

answered 1 hour ago

Dmitry KamenetskyDmitry Kamenetsky

96022 silver badges1818 bronze badges

$endgroup$

share|improve this answer

answered 1 hour ago

Dmitry KamenetskyDmitry Kamenetsky

96022 silver badges1818 bronze badges

answered 1 hour ago

Dmitry KamenetskyDmitry Kamenetsky

96022 silver badges1818 bronze badges

answered 1 hour ago

Dmitry KamenetskyDmitry Kamenetsky

96022 silver badges1818 bronze badges