How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$?...
$begingroup$
How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$?
Examples, for $n=3$, there is a twin prime between 9 and 25 of (11,13).
For $n=9$, there is a twin prime between 81 and 121 of (101,103).
number-theory prime-numbers prime-twins
asked Jan 18 at 21:14
temp wattstemp watts
1126
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closed as off-topic by Peter, José Carlos Santos, rtybase, mrtaurho, Leucippus Jan 20 at 1:47
This question appears to be off-topic. The users who voted to close gave these specific reasons:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – rtybase, Leucippus
- "This question is not about mathematics, within the scope defined in the help center." – Peter, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
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show 2 more comments
$begingroup$
How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$?
Examples, for $n=3$, there is a twin prime between 9 and 25 of (11,13).
For $n=9$, there is a twin prime between 81 and 121 of (101,103).
number-theory prime-numbers prime-twins
asked Jan 18 at 21:14
temp wattstemp watts
1126
$endgroup$
closed as off-topic by Peter, José Carlos Santos, rtybase, mrtaurho, Leucippus Jan 20 at 1:47
This question appears to be off-topic. The users who voted to close gave these specific reasons:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – rtybase, Leucippus
- "This question is not about mathematics, within the scope defined in the help center." – Peter, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
3
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Since it is even not known whether there are infinitely many twin primes, speculation on the methods of proof of something even stronger is likely dubious.
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– Mindlack
Jan 18 at 21:18
1
$begingroup$
Why would you imagine that this is easier than the Twin Prime Conjecture?
$endgroup$
– lulu
Jan 18 at 21:18
1
$begingroup$
Already, it is not easy at all to show that there is a prime between n and 2n (just a prime, not à twin prime)
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– DLeMeur
Jan 18 at 21:20
1
$begingroup$
You could prove it false by exhibiting an $n$ where it fails. That would not mean settling the twin prime conjecture because there could still be an infinite number of them, just a hole in the distribution.
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– Ross Millikan
Jan 19 at 2:12
1
$begingroup$
The statement is stronger than the twin-prime-conjecture, so it is very unlikely that someone can prove the conjecture.
$endgroup$
– Peter
Jan 19 at 14:05
|
show 2 more comments
$begingroup$
How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$?
Examples, for $n=3$, there is a twin prime between 9 and 25 of (11,13).
For $n=9$, there is a twin prime between 81 and 121 of (101,103).
number-theory prime-numbers prime-twins
asked Jan 18 at 21:14
temp wattstemp watts
1126
$endgroup$
How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$?
Examples, for $n=3$, there is a twin prime between 9 and 25 of (11,13).
For $n=9$, there is a twin prime between 81 and 121 of (101,103).
number-theory prime-numbers prime-twins
number-theory prime-numbers prime-twins
asked Jan 18 at 21:14
temp wattstemp watts
1126
asked Jan 18 at 21:14
temp wattstemp watts
1126
asked Jan 18 at 21:14
temp wattstemp watts
1126
asked Jan 18 at 21:14
temp wattstemp watts
1126
asked Jan 18 at 21:14
temp wattstemp watts
1126
1126
closed as off-topic by Peter, José Carlos Santos, rtybase, mrtaurho, Leucippus Jan 20 at 1:47
This question appears to be off-topic. The users who voted to close gave these specific reasons:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – rtybase, Leucippus
- "This question is not about mathematics, within the scope defined in the help center." – Peter, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Peter, José Carlos Santos, rtybase, mrtaurho, Leucippus Jan 20 at 1:47
This question appears to be off-topic. The users who voted to close gave these specific reasons:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – rtybase, Leucippus
- "This question is not about mathematics, within the scope defined in the help center." – Peter, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
3
$begingroup$
Since it is even not known whether there are infinitely many twin primes, speculation on the methods of proof of something even stronger is likely dubious.
$endgroup$
– Mindlack
Jan 18 at 21:18
1
$begingroup$
Why would you imagine that this is easier than the Twin Prime Conjecture?
$endgroup$
– lulu
Jan 18 at 21:18
1
$begingroup$
Already, it is not easy at all to show that there is a prime between n and 2n (just a prime, not à twin prime)
$endgroup$
– DLeMeur
Jan 18 at 21:20
1
$begingroup$
You could prove it false by exhibiting an $n$ where it fails. That would not mean settling the twin prime conjecture because there could still be an infinite number of them, just a hole in the distribution.
$endgroup$
– Ross Millikan
Jan 19 at 2:12
1
$begingroup$
The statement is stronger than the twin-prime-conjecture, so it is very unlikely that someone can prove the conjecture.
$endgroup$
– Peter
Jan 19 at 14:05
|
show 2 more comments
3
$begingroup$
Since it is even not known whether there are infinitely many twin primes, speculation on the methods of proof of something even stronger is likely dubious.
$endgroup$
– Mindlack
Jan 18 at 21:18
1
$begingroup$
Why would you imagine that this is easier than the Twin Prime Conjecture?
$endgroup$
– lulu
Jan 18 at 21:18
1
$begingroup$
Already, it is not easy at all to show that there is a prime between n and 2n (just a prime, not à twin prime)
$endgroup$
– DLeMeur
Jan 18 at 21:20
1
$begingroup$
You could prove it false by exhibiting an $n$ where it fails. That would not mean settling the twin prime conjecture because there could still be an infinite number of them, just a hole in the distribution.
$endgroup$
– Ross Millikan
Jan 19 at 2:12
1
$begingroup$
The statement is stronger than the twin-prime-conjecture, so it is very unlikely that someone can prove the conjecture.
$endgroup$
– Peter
Jan 19 at 14:05
3
3
$begingroup$
Since it is even not known whether there are infinitely many twin primes, speculation on the methods of proof of something even stronger is likely dubious.
$endgroup$
– Mindlack
Jan 18 at 21:18
$begingroup$
Since it is even not known whether there are infinitely many twin primes, speculation on the methods of proof of something even stronger is likely dubious.
$endgroup$
– Mindlack
Jan 18 at 21:18
1
1
$begingroup$
Why would you imagine that this is easier than the Twin Prime Conjecture?
$endgroup$
– lulu
Jan 18 at 21:18
$begingroup$
Why would you imagine that this is easier than the Twin Prime Conjecture?
$endgroup$
– lulu
Jan 18 at 21:18
1
1
$begingroup$
Already, it is not easy at all to show that there is a prime between n and 2n (just a prime, not à twin prime)
$endgroup$
– DLeMeur
Jan 18 at 21:20
$begingroup$
Already, it is not easy at all to show that there is a prime between n and 2n (just a prime, not à twin prime)
$endgroup$
– DLeMeur
Jan 18 at 21:20
1
1
$begingroup$
You could prove it false by exhibiting an $n$ where it fails. That would not mean settling the twin prime conjecture because there could still be an infinite number of them, just a hole in the distribution.
$endgroup$
– Ross Millikan
Jan 19 at 2:12
$begingroup$
You could prove it false by exhibiting an $n$ where it fails. That would not mean settling the twin prime conjecture because there could still be an infinite number of them, just a hole in the distribution.
$endgroup$
– Ross Millikan
Jan 19 at 2:12
1
1
$begingroup$
The statement is stronger than the twin-prime-conjecture, so it is very unlikely that someone can prove the conjecture.
$endgroup$
– Peter
Jan 19 at 14:05
$begingroup$
The statement is stronger than the twin-prime-conjecture, so it is very unlikely that someone can prove the conjecture.
$endgroup$
– Peter
Jan 19 at 14:05
|
show 2 more comments
2 Answers
2
active
oldest
votes
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Out of curiosity, I wrote & ran a fairly simple program to check certain information regarding how many twin primes there are between $n^2$ and $left(n + 2right)^2$ for odd natural numbers $n$. I ran this for $n$ up to a million, i.e., $100000$, so the square goes up to a trillion. For each range of billion integers (apart from the last one due to a small coding limitation), I output the cumulative minimum, maximum & average of the # of twin primes in each $n^2$ to $left(n + 2right)^2$ range. The minimum value of $1$ first occurs for $n = 21$, but I didn't check if it occurred again afterwards. The maximum value goes up for each billion initially, but then sometimes doesn't change for a span of quite a few billion. At the end, it is $7207$. The average seems to always be increasing fairly steadily, but slower later on. At the end, it is about $3739.588515$.
This indicates your conjecture appears to be plausible, but it's obviously not a proof. As Mostafa Ayaz states, the twin prime conjecture is not proven so we don't even know for sure that there's an infinite # of twin primes, much less at least one between each $n^2$ and $left(n + 2right)^2$ range.
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
$endgroup$
$begingroup$
Thank you very much! You went well beyond what I was expecting for an answer. Since it is a plausible conjecture but unsolvable, can I call it the "Watanabe Conjecture" and maybe it will become a famous conjecture like the Goldbach conjecture?
$endgroup$
– temp watts
Jan 20 at 21:43
$begingroup$
@tempwatts You are welcome. You can certainly call it whatever you want and, you never know, it may become a famous conjecture some day.
$endgroup$
– John Omielan
Jan 20 at 21:45
add a comment |
$begingroup$
You can't.
According to Twin primes on Wikipedia, the number of twin prime couples is unknown (even whether they are finite or infinite). If your statement holds true, then as a consequence you have proved that there are infinitely many twin primes which is yet unsolved. Your question then sounds very encouraging as a rush to solve this old and interesting conjecture!
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Out of curiosity, I wrote & ran a fairly simple program to check certain information regarding how many twin primes there are between $n^2$ and $left(n + 2right)^2$ for odd natural numbers $n$. I ran this for $n$ up to a million, i.e., $100000$, so the square goes up to a trillion. For each range of billion integers (apart from the last one due to a small coding limitation), I output the cumulative minimum, maximum & average of the # of twin primes in each $n^2$ to $left(n + 2right)^2$ range. The minimum value of $1$ first occurs for $n = 21$, but I didn't check if it occurred again afterwards. The maximum value goes up for each billion initially, but then sometimes doesn't change for a span of quite a few billion. At the end, it is $7207$. The average seems to always be increasing fairly steadily, but slower later on. At the end, it is about $3739.588515$.
This indicates your conjecture appears to be plausible, but it's obviously not a proof. As Mostafa Ayaz states, the twin prime conjecture is not proven so we don't even know for sure that there's an infinite # of twin primes, much less at least one between each $n^2$ and $left(n + 2right)^2$ range.
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
$endgroup$
$begingroup$
Thank you very much! You went well beyond what I was expecting for an answer. Since it is a plausible conjecture but unsolvable, can I call it the "Watanabe Conjecture" and maybe it will become a famous conjecture like the Goldbach conjecture?
$endgroup$
– temp watts
Jan 20 at 21:43
$begingroup$
@tempwatts You are welcome. You can certainly call it whatever you want and, you never know, it may become a famous conjecture some day.
$endgroup$
– John Omielan
Jan 20 at 21:45
add a comment |
$begingroup$
Out of curiosity, I wrote & ran a fairly simple program to check certain information regarding how many twin primes there are between $n^2$ and $left(n + 2right)^2$ for odd natural numbers $n$. I ran this for $n$ up to a million, i.e., $100000$, so the square goes up to a trillion. For each range of billion integers (apart from the last one due to a small coding limitation), I output the cumulative minimum, maximum & average of the # of twin primes in each $n^2$ to $left(n + 2right)^2$ range. The minimum value of $1$ first occurs for $n = 21$, but I didn't check if it occurred again afterwards. The maximum value goes up for each billion initially, but then sometimes doesn't change for a span of quite a few billion. At the end, it is $7207$. The average seems to always be increasing fairly steadily, but slower later on. At the end, it is about $3739.588515$.
This indicates your conjecture appears to be plausible, but it's obviously not a proof. As Mostafa Ayaz states, the twin prime conjecture is not proven so we don't even know for sure that there's an infinite # of twin primes, much less at least one between each $n^2$ and $left(n + 2right)^2$ range.
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
$endgroup$
$begingroup$
Thank you very much! You went well beyond what I was expecting for an answer. Since it is a plausible conjecture but unsolvable, can I call it the "Watanabe Conjecture" and maybe it will become a famous conjecture like the Goldbach conjecture?
$endgroup$
– temp watts
Jan 20 at 21:43
$begingroup$
@tempwatts You are welcome. You can certainly call it whatever you want and, you never know, it may become a famous conjecture some day.
$endgroup$
– John Omielan
Jan 20 at 21:45
add a comment |
$begingroup$
Out of curiosity, I wrote & ran a fairly simple program to check certain information regarding how many twin primes there are between $n^2$ and $left(n + 2right)^2$ for odd natural numbers $n$. I ran this for $n$ up to a million, i.e., $100000$, so the square goes up to a trillion. For each range of billion integers (apart from the last one due to a small coding limitation), I output the cumulative minimum, maximum & average of the # of twin primes in each $n^2$ to $left(n + 2right)^2$ range. The minimum value of $1$ first occurs for $n = 21$, but I didn't check if it occurred again afterwards. The maximum value goes up for each billion initially, but then sometimes doesn't change for a span of quite a few billion. At the end, it is $7207$. The average seems to always be increasing fairly steadily, but slower later on. At the end, it is about $3739.588515$.
This indicates your conjecture appears to be plausible, but it's obviously not a proof. As Mostafa Ayaz states, the twin prime conjecture is not proven so we don't even know for sure that there's an infinite # of twin primes, much less at least one between each $n^2$ and $left(n + 2right)^2$ range.
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
$endgroup$
Out of curiosity, I wrote & ran a fairly simple program to check certain information regarding how many twin primes there are between $n^2$ and $left(n + 2right)^2$ for odd natural numbers $n$. I ran this for $n$ up to a million, i.e., $100000$, so the square goes up to a trillion. For each range of billion integers (apart from the last one due to a small coding limitation), I output the cumulative minimum, maximum & average of the # of twin primes in each $n^2$ to $left(n + 2right)^2$ range. The minimum value of $1$ first occurs for $n = 21$, but I didn't check if it occurred again afterwards. The maximum value goes up for each billion initially, but then sometimes doesn't change for a span of quite a few billion. At the end, it is $7207$. The average seems to always be increasing fairly steadily, but slower later on. At the end, it is about $3739.588515$.
This indicates your conjecture appears to be plausible, but it's obviously not a proof. As Mostafa Ayaz states, the twin prime conjecture is not proven so we don't even know for sure that there's an infinite # of twin primes, much less at least one between each $n^2$ and $left(n + 2right)^2$ range.
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
answered Jan 19 at 1:56
John OmielanJohn Omielan
5,2542218
5,2542218
$begingroup$
Thank you very much! You went well beyond what I was expecting for an answer. Since it is a plausible conjecture but unsolvable, can I call it the "Watanabe Conjecture" and maybe it will become a famous conjecture like the Goldbach conjecture?
$endgroup$
– temp watts
Jan 20 at 21:43
$begingroup$
@tempwatts You are welcome. You can certainly call it whatever you want and, you never know, it may become a famous conjecture some day.
$endgroup$
– John Omielan
Jan 20 at 21:45
add a comment |
$begingroup$
Thank you very much! You went well beyond what I was expecting for an answer. Since it is a plausible conjecture but unsolvable, can I call it the "Watanabe Conjecture" and maybe it will become a famous conjecture like the Goldbach conjecture?
$endgroup$
– temp watts
Jan 20 at 21:43
$begingroup$
@tempwatts You are welcome. You can certainly call it whatever you want and, you never know, it may become a famous conjecture some day.
$endgroup$
– John Omielan
Jan 20 at 21:45
$begingroup$
Thank you very much! You went well beyond what I was expecting for an answer. Since it is a plausible conjecture but unsolvable, can I call it the "Watanabe Conjecture" and maybe it will become a famous conjecture like the Goldbach conjecture?
$endgroup$
– temp watts
Jan 20 at 21:43
$begingroup$
Thank you very much! You went well beyond what I was expecting for an answer. Since it is a plausible conjecture but unsolvable, can I call it the "Watanabe Conjecture" and maybe it will become a famous conjecture like the Goldbach conjecture?
$endgroup$
– temp watts
Jan 20 at 21:43
$begingroup$
@tempwatts You are welcome. You can certainly call it whatever you want and, you never know, it may become a famous conjecture some day.
$endgroup$
– John Omielan
Jan 20 at 21:45
$begingroup$
@tempwatts You are welcome. You can certainly call it whatever you want and, you never know, it may become a famous conjecture some day.
$endgroup$
– John Omielan
Jan 20 at 21:45
add a comment |
$begingroup$
You can't.
According to Twin primes on Wikipedia, the number of twin prime couples is unknown (even whether they are finite or infinite). If your statement holds true, then as a consequence you have proved that there are infinitely many twin primes which is yet unsolved. Your question then sounds very encouraging as a rush to solve this old and interesting conjecture!
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
add a comment |
$begingroup$
You can't.
According to Twin primes on Wikipedia, the number of twin prime couples is unknown (even whether they are finite or infinite). If your statement holds true, then as a consequence you have proved that there are infinitely many twin primes which is yet unsolved. Your question then sounds very encouraging as a rush to solve this old and interesting conjecture!
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
add a comment |
$begingroup$
You can't.
According to Twin primes on Wikipedia, the number of twin prime couples is unknown (even whether they are finite or infinite). If your statement holds true, then as a consequence you have proved that there are infinitely many twin primes which is yet unsolved. Your question then sounds very encouraging as a rush to solve this old and interesting conjecture!
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
You can't.
According to Twin primes on Wikipedia, the number of twin prime couples is unknown (even whether they are finite or infinite). If your statement holds true, then as a consequence you have proved that there are infinitely many twin primes which is yet unsolved. Your question then sounds very encouraging as a rush to solve this old and interesting conjecture!
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
answered Jan 18 at 21:33
Mostafa AyazMostafa Ayaz
18.1k31040
18.1k31040
add a comment |
add a comment |
3
$begingroup$
Since it is even not known whether there are infinitely many twin primes, speculation on the methods of proof of something even stronger is likely dubious.
$endgroup$
– Mindlack
Jan 18 at 21:18
1
$begingroup$
Why would you imagine that this is easier than the Twin Prime Conjecture?
$endgroup$
– lulu
Jan 18 at 21:18
1
$begingroup$
Already, it is not easy at all to show that there is a prime between n and 2n (just a prime, not à twin prime)
$endgroup$
– DLeMeur
Jan 18 at 21:20
1
$begingroup$
You could prove it false by exhibiting an $n$ where it fails. That would not mean settling the twin prime conjecture because there could still be an infinite number of them, just a hole in the distribution.
$endgroup$
– Ross Millikan
Jan 19 at 2:12
1
$begingroup$
The statement is stronger than the twin-prime-conjecture, so it is very unlikely that someone can prove the conjecture.
$endgroup$
– Peter
Jan 19 at 14:05