Wieferich primes in base $47$
Wieferich primes are defined as prime numbers $p$ such that $p^2$ divides $2^{p − 1} − 1$. While reading about such primes, I came upon the following curious conjecture on the Wikipedia page of "unsolved problems in number theory":
Are there any Wieferich primes in base $47$?
Since no explanation is given for this strange question, I find myself puzzled by the importance of the number $47$ within this context. What role does this base in particular have in the context of Wieferich primes and why would it be important to solve this problem in particular, instead of another number base?
elementary-number-theory prime-numbers
asked Apr 4 '18 at 12:11
Klangen
1,65711334
|
show 4 more comments
Wieferich primes are defined as prime numbers $p$ such that $p^2$ divides $2^{p − 1} − 1$. While reading about such primes, I came upon the following curious conjecture on the Wikipedia page of "unsolved problems in number theory":
Are there any Wieferich primes in base $47$?
Since no explanation is given for this strange question, I find myself puzzled by the importance of the number $47$ within this context. What role does this base in particular have in the context of Wieferich primes and why would it be important to solve this problem in particular, instead of another number base?
elementary-number-theory prime-numbers
asked Apr 4 '18 at 12:11
Klangen
1,65711334
3
Perhaps someone looked, and found Wieferich primes in all other small bases.
– Michael
Apr 4 '18 at 12:15
2
In fact, $47$ is the smallest base without a known example.
– Peter
Apr 4 '18 at 12:16
1
What does the base even have to do with it?
– vrugtehagel
Apr 4 '18 at 12:20
1
See this
– steven gregory
Apr 4 '18 at 12:22
Yup, just found the relevant bit in the wikipedia article. Thanks anyway :)
– vrugtehagel
Apr 4 '18 at 12:22
|
show 4 more comments
Wieferich primes are defined as prime numbers $p$ such that $p^2$ divides $2^{p − 1} − 1$. While reading about such primes, I came upon the following curious conjecture on the Wikipedia page of "unsolved problems in number theory":
Are there any Wieferich primes in base $47$?
Since no explanation is given for this strange question, I find myself puzzled by the importance of the number $47$ within this context. What role does this base in particular have in the context of Wieferich primes and why would it be important to solve this problem in particular, instead of another number base?
elementary-number-theory prime-numbers
asked Apr 4 '18 at 12:11
Klangen
1,65711334
Wieferich primes are defined as prime numbers $p$ such that $p^2$ divides $2^{p − 1} − 1$. While reading about such primes, I came upon the following curious conjecture on the Wikipedia page of "unsolved problems in number theory":
Are there any Wieferich primes in base $47$?
Since no explanation is given for this strange question, I find myself puzzled by the importance of the number $47$ within this context. What role does this base in particular have in the context of Wieferich primes and why would it be important to solve this problem in particular, instead of another number base?
elementary-number-theory prime-numbers
elementary-number-theory prime-numbers
asked Apr 4 '18 at 12:11
Klangen
1,65711334
asked Apr 4 '18 at 12:11
Klangen
1,65711334
asked Apr 4 '18 at 12:11
Klangen
1,65711334
asked Apr 4 '18 at 12:11
Klangen
1,65711334
asked Apr 4 '18 at 12:11
Klangen
1,65711334
1,65711334
3
Perhaps someone looked, and found Wieferich primes in all other small bases.
– Michael
Apr 4 '18 at 12:15
2
In fact, $47$ is the smallest base without a known example.
– Peter
Apr 4 '18 at 12:16
1
What does the base even have to do with it?
– vrugtehagel
Apr 4 '18 at 12:20
1
See this
– steven gregory
Apr 4 '18 at 12:22
Yup, just found the relevant bit in the wikipedia article. Thanks anyway :)
– vrugtehagel
Apr 4 '18 at 12:22
|
show 4 more comments
3
Perhaps someone looked, and found Wieferich primes in all other small bases.
– Michael
Apr 4 '18 at 12:15
2
In fact, $47$ is the smallest base without a known example.
– Peter
Apr 4 '18 at 12:16
1
What does the base even have to do with it?
– vrugtehagel
Apr 4 '18 at 12:20
1
See this
– steven gregory
Apr 4 '18 at 12:22
Yup, just found the relevant bit in the wikipedia article. Thanks anyway :)
– vrugtehagel
Apr 4 '18 at 12:22
3
3
Perhaps someone looked, and found Wieferich primes in all other small bases.
– Michael
Apr 4 '18 at 12:15
Perhaps someone looked, and found Wieferich primes in all other small bases.
– Michael
Apr 4 '18 at 12:15
2
2
In fact, $47$ is the smallest base without a known example.
– Peter
Apr 4 '18 at 12:16
In fact, $47$ is the smallest base without a known example.
– Peter
Apr 4 '18 at 12:16
1
1
What does the base even have to do with it?
– vrugtehagel
Apr 4 '18 at 12:20
What does the base even have to do with it?
– vrugtehagel
Apr 4 '18 at 12:20
1
1
See this
– steven gregory
Apr 4 '18 at 12:22
See this
– steven gregory
Apr 4 '18 at 12:22
Yup, just found the relevant bit in the wikipedia article. Thanks anyway :)
– vrugtehagel
Apr 4 '18 at 12:22
Yup, just found the relevant bit in the wikipedia article. Thanks anyway :)
– vrugtehagel
Apr 4 '18 at 12:22
|
show 4 more comments
2 Answers
2
active
oldest
votes
After some research on the internet, it indeed seems that Peter was right in the comment section, and $47$ is the smallest base for which no Wieferich primes are known.
answered yesterday
Klangen
1,65711334
add a comment |
Weiferich primes exist only in base 2. He wrote his paper in 1908, but the question has been around since the time of Euler. Dickson's history of mathematics devotes 12 pages to this question, but affords Weiferich only five lines in a paragraph on the nineth page. It is misleading to use this term.
Weiferich did not discover sevenites, but he noted that a particular solution to fermat's last proposition exists only for binary sevenites.
There is no particular reason for 47 not to have any particular sevenites. A similar situation exists with there being no known iso-sevenites for 3, (the Fibonacci series) which means that no instance of where if $p mid F_n$ then also $p^2 mid F_n$. However, the octagon-series and the Heron series, which correspond to isobases 6 and 4, do have sevenites.
Thus, in the series of Heron triangles, (triangles of sides e-1, e, e+1 and integer area), if 103 divides a side, so does $103^2$.
47 has 2 as a sevenite (or 'weiferich prime'). The two-place period of 2 supposes only 8, as can be seen in 11, 13 and 45. Here 32 divides 47^2-1, and thus it has two as a sevenite.
EDIT:
The table of 'sevenites' for particular bases, up to b=14400 and p=2000000, do not produce a list of primes longer than 80 digits, except in one or two cases. The number is quite small. Sort of in the $sum 1/p$ range. There are a good scattering of unfilled rows, 47 is the first.
In the early versions of the tables I produced 6 was the first unfilled row. But then 61661 came along.
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
2
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review
– Leucippus
Apr 4 '18 at 15:39
It actually does. The iteration beginning 2,b, and continuing t(n+1)=b.t(n)-t(n-1), is a very base-like structure, and the distribution of p²|t(p)-b, is identical to that if p²|b^p-b. The example of b=3 gives the lucas numbers.
– wendy.krieger
Apr 5 '18 at 7:10
I have not been able to find any reference to the general class of sevenite being called Weiferich numbers, until i posted these as sevenites on the dozenal list as an entity, and D.S. reusing the name 'Weiferich' as a general name. I stick by my name.
– wendy.krieger
Apr 5 '18 at 7:12
1
11 is a Wieferich prime base 3 because 3^10 = 1 modulo 11^2, so they CAN exists in other bases not 2
– J. Linne
Apr 14 '18 at 5:23
@J.Linne The two wieferich primes are 1093 and 3511. In base 3, this is 1111111 and 11211001. Sevenites in base 3, such as 11, are Marianoff primes. Decimal sevenites such as 3 and 487 are Shanks Primes. I studied the thing back in the eighties, when the general member had no name. I called them sevenites.
– wendy.krieger
Apr 14 '18 at 9:45
|
show 1 more comment
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2 Answers
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2 Answers
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After some research on the internet, it indeed seems that Peter was right in the comment section, and $47$ is the smallest base for which no Wieferich primes are known.
answered yesterday
Klangen
1,65711334
add a comment |
After some research on the internet, it indeed seems that Peter was right in the comment section, and $47$ is the smallest base for which no Wieferich primes are known.
answered yesterday
Klangen
1,65711334
add a comment |
After some research on the internet, it indeed seems that Peter was right in the comment section, and $47$ is the smallest base for which no Wieferich primes are known.
answered yesterday
Klangen
1,65711334
After some research on the internet, it indeed seems that Peter was right in the comment section, and $47$ is the smallest base for which no Wieferich primes are known.
answered yesterday
Klangen
1,65711334
answered yesterday
Klangen
1,65711334
answered yesterday
Klangen
1,65711334
answered yesterday
Klangen
1,65711334
1,65711334
add a comment |
add a comment |
Weiferich primes exist only in base 2. He wrote his paper in 1908, but the question has been around since the time of Euler. Dickson's history of mathematics devotes 12 pages to this question, but affords Weiferich only five lines in a paragraph on the nineth page. It is misleading to use this term.
Weiferich did not discover sevenites, but he noted that a particular solution to fermat's last proposition exists only for binary sevenites.
There is no particular reason for 47 not to have any particular sevenites. A similar situation exists with there being no known iso-sevenites for 3, (the Fibonacci series) which means that no instance of where if $p mid F_n$ then also $p^2 mid F_n$. However, the octagon-series and the Heron series, which correspond to isobases 6 and 4, do have sevenites.
Thus, in the series of Heron triangles, (triangles of sides e-1, e, e+1 and integer area), if 103 divides a side, so does $103^2$.
47 has 2 as a sevenite (or 'weiferich prime'). The two-place period of 2 supposes only 8, as can be seen in 11, 13 and 45. Here 32 divides 47^2-1, and thus it has two as a sevenite.
EDIT:
The table of 'sevenites' for particular bases, up to b=14400 and p=2000000, do not produce a list of primes longer than 80 digits, except in one or two cases. The number is quite small. Sort of in the $sum 1/p$ range. There are a good scattering of unfilled rows, 47 is the first.
In the early versions of the tables I produced 6 was the first unfilled row. But then 61661 came along.
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
2
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review
– Leucippus
Apr 4 '18 at 15:39
It actually does. The iteration beginning 2,b, and continuing t(n+1)=b.t(n)-t(n-1), is a very base-like structure, and the distribution of p²|t(p)-b, is identical to that if p²|b^p-b. The example of b=3 gives the lucas numbers.
– wendy.krieger
Apr 5 '18 at 7:10
I have not been able to find any reference to the general class of sevenite being called Weiferich numbers, until i posted these as sevenites on the dozenal list as an entity, and D.S. reusing the name 'Weiferich' as a general name. I stick by my name.
– wendy.krieger
Apr 5 '18 at 7:12
1
11 is a Wieferich prime base 3 because 3^10 = 1 modulo 11^2, so they CAN exists in other bases not 2
– J. Linne
Apr 14 '18 at 5:23
@J.Linne The two wieferich primes are 1093 and 3511. In base 3, this is 1111111 and 11211001. Sevenites in base 3, such as 11, are Marianoff primes. Decimal sevenites such as 3 and 487 are Shanks Primes. I studied the thing back in the eighties, when the general member had no name. I called them sevenites.
– wendy.krieger
Apr 14 '18 at 9:45
|
show 1 more comment
Weiferich primes exist only in base 2. He wrote his paper in 1908, but the question has been around since the time of Euler. Dickson's history of mathematics devotes 12 pages to this question, but affords Weiferich only five lines in a paragraph on the nineth page. It is misleading to use this term.
Weiferich did not discover sevenites, but he noted that a particular solution to fermat's last proposition exists only for binary sevenites.
There is no particular reason for 47 not to have any particular sevenites. A similar situation exists with there being no known iso-sevenites for 3, (the Fibonacci series) which means that no instance of where if $p mid F_n$ then also $p^2 mid F_n$. However, the octagon-series and the Heron series, which correspond to isobases 6 and 4, do have sevenites.
Thus, in the series of Heron triangles, (triangles of sides e-1, e, e+1 and integer area), if 103 divides a side, so does $103^2$.
47 has 2 as a sevenite (or 'weiferich prime'). The two-place period of 2 supposes only 8, as can be seen in 11, 13 and 45. Here 32 divides 47^2-1, and thus it has two as a sevenite.
EDIT:
The table of 'sevenites' for particular bases, up to b=14400 and p=2000000, do not produce a list of primes longer than 80 digits, except in one or two cases. The number is quite small. Sort of in the $sum 1/p$ range. There are a good scattering of unfilled rows, 47 is the first.
In the early versions of the tables I produced 6 was the first unfilled row. But then 61661 came along.
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
2
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review
– Leucippus
Apr 4 '18 at 15:39
It actually does. The iteration beginning 2,b, and continuing t(n+1)=b.t(n)-t(n-1), is a very base-like structure, and the distribution of p²|t(p)-b, is identical to that if p²|b^p-b. The example of b=3 gives the lucas numbers.
– wendy.krieger
Apr 5 '18 at 7:10
I have not been able to find any reference to the general class of sevenite being called Weiferich numbers, until i posted these as sevenites on the dozenal list as an entity, and D.S. reusing the name 'Weiferich' as a general name. I stick by my name.
– wendy.krieger
Apr 5 '18 at 7:12
1
11 is a Wieferich prime base 3 because 3^10 = 1 modulo 11^2, so they CAN exists in other bases not 2
– J. Linne
Apr 14 '18 at 5:23
@J.Linne The two wieferich primes are 1093 and 3511. In base 3, this is 1111111 and 11211001. Sevenites in base 3, such as 11, are Marianoff primes. Decimal sevenites such as 3 and 487 are Shanks Primes. I studied the thing back in the eighties, when the general member had no name. I called them sevenites.
– wendy.krieger
Apr 14 '18 at 9:45
|
show 1 more comment
Weiferich primes exist only in base 2. He wrote his paper in 1908, but the question has been around since the time of Euler. Dickson's history of mathematics devotes 12 pages to this question, but affords Weiferich only five lines in a paragraph on the nineth page. It is misleading to use this term.
Weiferich did not discover sevenites, but he noted that a particular solution to fermat's last proposition exists only for binary sevenites.
There is no particular reason for 47 not to have any particular sevenites. A similar situation exists with there being no known iso-sevenites for 3, (the Fibonacci series) which means that no instance of where if $p mid F_n$ then also $p^2 mid F_n$. However, the octagon-series and the Heron series, which correspond to isobases 6 and 4, do have sevenites.
Thus, in the series of Heron triangles, (triangles of sides e-1, e, e+1 and integer area), if 103 divides a side, so does $103^2$.
47 has 2 as a sevenite (or 'weiferich prime'). The two-place period of 2 supposes only 8, as can be seen in 11, 13 and 45. Here 32 divides 47^2-1, and thus it has two as a sevenite.
EDIT:
The table of 'sevenites' for particular bases, up to b=14400 and p=2000000, do not produce a list of primes longer than 80 digits, except in one or two cases. The number is quite small. Sort of in the $sum 1/p$ range. There are a good scattering of unfilled rows, 47 is the first.
In the early versions of the tables I produced 6 was the first unfilled row. But then 61661 came along.
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
Weiferich primes exist only in base 2. He wrote his paper in 1908, but the question has been around since the time of Euler. Dickson's history of mathematics devotes 12 pages to this question, but affords Weiferich only five lines in a paragraph on the nineth page. It is misleading to use this term.
Weiferich did not discover sevenites, but he noted that a particular solution to fermat's last proposition exists only for binary sevenites.
There is no particular reason for 47 not to have any particular sevenites. A similar situation exists with there being no known iso-sevenites for 3, (the Fibonacci series) which means that no instance of where if $p mid F_n$ then also $p^2 mid F_n$. However, the octagon-series and the Heron series, which correspond to isobases 6 and 4, do have sevenites.
Thus, in the series of Heron triangles, (triangles of sides e-1, e, e+1 and integer area), if 103 divides a side, so does $103^2$.
47 has 2 as a sevenite (or 'weiferich prime'). The two-place period of 2 supposes only 8, as can be seen in 11, 13 and 45. Here 32 divides 47^2-1, and thus it has two as a sevenite.
EDIT:
The table of 'sevenites' for particular bases, up to b=14400 and p=2000000, do not produce a list of primes longer than 80 digits, except in one or two cases. The number is quite small. Sort of in the $sum 1/p$ range. There are a good scattering of unfilled rows, 47 is the first.
In the early versions of the tables I produced 6 was the first unfilled row. But then 61661 came along.
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
edited yesterday
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
answered Apr 4 '18 at 12:23
wendy.krieger
5,74111426
5,74111426
2
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review
– Leucippus
Apr 4 '18 at 15:39
It actually does. The iteration beginning 2,b, and continuing t(n+1)=b.t(n)-t(n-1), is a very base-like structure, and the distribution of p²|t(p)-b, is identical to that if p²|b^p-b. The example of b=3 gives the lucas numbers.
– wendy.krieger
Apr 5 '18 at 7:10
I have not been able to find any reference to the general class of sevenite being called Weiferich numbers, until i posted these as sevenites on the dozenal list as an entity, and D.S. reusing the name 'Weiferich' as a general name. I stick by my name.
– wendy.krieger
Apr 5 '18 at 7:12
1
11 is a Wieferich prime base 3 because 3^10 = 1 modulo 11^2, so they CAN exists in other bases not 2
– J. Linne
Apr 14 '18 at 5:23
@J.Linne The two wieferich primes are 1093 and 3511. In base 3, this is 1111111 and 11211001. Sevenites in base 3, such as 11, are Marianoff primes. Decimal sevenites such as 3 and 487 are Shanks Primes. I studied the thing back in the eighties, when the general member had no name. I called them sevenites.
– wendy.krieger
Apr 14 '18 at 9:45
|
show 1 more comment
2
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review
– Leucippus
Apr 4 '18 at 15:39
It actually does. The iteration beginning 2,b, and continuing t(n+1)=b.t(n)-t(n-1), is a very base-like structure, and the distribution of p²|t(p)-b, is identical to that if p²|b^p-b. The example of b=3 gives the lucas numbers.
– wendy.krieger
Apr 5 '18 at 7:10
I have not been able to find any reference to the general class of sevenite being called Weiferich numbers, until i posted these as sevenites on the dozenal list as an entity, and D.S. reusing the name 'Weiferich' as a general name. I stick by my name.
– wendy.krieger
Apr 5 '18 at 7:12
1
11 is a Wieferich prime base 3 because 3^10 = 1 modulo 11^2, so they CAN exists in other bases not 2
– J. Linne
Apr 14 '18 at 5:23
@J.Linne The two wieferich primes are 1093 and 3511. In base 3, this is 1111111 and 11211001. Sevenites in base 3, such as 11, are Marianoff primes. Decimal sevenites such as 3 and 487 are Shanks Primes. I studied the thing back in the eighties, when the general member had no name. I called them sevenites.
– wendy.krieger
Apr 14 '18 at 9:45
2
2
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review
– Leucippus
Apr 4 '18 at 15:39
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review
– Leucippus
Apr 4 '18 at 15:39
It actually does. The iteration beginning 2,b, and continuing t(n+1)=b.t(n)-t(n-1), is a very base-like structure, and the distribution of p²|t(p)-b, is identical to that if p²|b^p-b. The example of b=3 gives the lucas numbers.
– wendy.krieger
Apr 5 '18 at 7:10
It actually does. The iteration beginning 2,b, and continuing t(n+1)=b.t(n)-t(n-1), is a very base-like structure, and the distribution of p²|t(p)-b, is identical to that if p²|b^p-b. The example of b=3 gives the lucas numbers.
– wendy.krieger
Apr 5 '18 at 7:10
I have not been able to find any reference to the general class of sevenite being called Weiferich numbers, until i posted these as sevenites on the dozenal list as an entity, and D.S. reusing the name 'Weiferich' as a general name. I stick by my name.
– wendy.krieger
Apr 5 '18 at 7:12
I have not been able to find any reference to the general class of sevenite being called Weiferich numbers, until i posted these as sevenites on the dozenal list as an entity, and D.S. reusing the name 'Weiferich' as a general name. I stick by my name.
– wendy.krieger
Apr 5 '18 at 7:12
1
1
11 is a Wieferich prime base 3 because 3^10 = 1 modulo 11^2, so they CAN exists in other bases not 2
– J. Linne
Apr 14 '18 at 5:23
11 is a Wieferich prime base 3 because 3^10 = 1 modulo 11^2, so they CAN exists in other bases not 2
– J. Linne
Apr 14 '18 at 5:23
@J.Linne The two wieferich primes are 1093 and 3511. In base 3, this is 1111111 and 11211001. Sevenites in base 3, such as 11, are Marianoff primes. Decimal sevenites such as 3 and 487 are Shanks Primes. I studied the thing back in the eighties, when the general member had no name. I called them sevenites.
– wendy.krieger
Apr 14 '18 at 9:45
@J.Linne The two wieferich primes are 1093 and 3511. In base 3, this is 1111111 and 11211001. Sevenites in base 3, such as 11, are Marianoff primes. Decimal sevenites such as 3 and 487 are Shanks Primes. I studied the thing back in the eighties, when the general member had no name. I called them sevenites.
– wendy.krieger
Apr 14 '18 at 9:45
|
show 1 more comment
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3
Perhaps someone looked, and found Wieferich primes in all other small bases.
– Michael
Apr 4 '18 at 12:15
2
In fact, $47$ is the smallest base without a known example.
– Peter
Apr 4 '18 at 12:16
1
What does the base even have to do with it?
– vrugtehagel
Apr 4 '18 at 12:20
1
See this
– steven gregory
Apr 4 '18 at 12:22
Yup, just found the relevant bit in the wikipedia article. Thanks anyway :)
– vrugtehagel
Apr 4 '18 at 12:22