Why do all primes $n>3$ satisfy $,309mid 20^n-13^n-7^n$ Unicorn Meta Zoo #1: Why another...
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Why do all primes $n>3$ satisfy $,309mid 20^n-13^n-7^n$
Unicorn Meta Zoo #1: Why another podcast?
Announcing the arrival of Valued Associate #679: Cesar ManaraPrime factor of $A=14^7+14^2+1$Number of divisors excluding set of primes SCalculating $e^{3x} pmod{27}$Help understanding number theory definitionWhat's wrong with this proof of the infinity of primes?Unclear on why Meissel's approach to counting primes worksIf $a^n+n^bmid c^n+n^d$ for every $n$ then $c=a^k$ and $d=kb$ .What does x equivalent to 2 mod 15 mean?Olympiad problem: Prove that there are infinitely many primes which divide the sequence ${ S(k) } _{kin mathbb{N}}$Primes of Atlantis: a definition and a problemWhat is an irreducible prime?
$begingroup$
Solve the following... $309|(20^n-13^n-7^n)$ in $mathbb{Z}^+$.
I invested lotof time to it and finally went to WolframAlpha for help by typing...
Solve $309k=20^n-13^n-7^n$ over the integers. It returned the following... Note that all the primes are generated here. Can someone please explain why? Thanks! EDIT: the formulas suggested in the answerare too complicated, but this is so simple!
elementary-number-theory
New contributor
$endgroup$
|
show 2 more comments
$begingroup$
Solve the following... $309|(20^n-13^n-7^n)$ in $mathbb{Z}^+$.
I invested lotof time to it and finally went to WolframAlpha for help by typing...
Solve $309k=20^n-13^n-7^n$ over the integers. It returned the following... Note that all the primes are generated here. Can someone please explain why? Thanks! EDIT: the formulas suggested in the answerare too complicated, but this is so simple!
elementary-number-theory
New contributor
$endgroup$
$begingroup$
Is it false that it generates all the primes?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
It is easy to picture why there might be a solution for all prime $n$. However, the interesting part of the pattern is that $n=5cdot 5$ and $n=5cdot 7$ both appear, yet neither $n=3cdot 3$ nor $n=5cdot 5$ appear. Which odd composites admit solutions?
$endgroup$
– Mark Fischler
11 hours ago
$begingroup$
There are no even numbers ..
$endgroup$
– Shamim Akhtar
10 hours ago
3
$begingroup$
It seems like $n equiv pm1 pmod 6$.
$endgroup$
– md2perpe
10 hours ago
$begingroup$
Yeah right but why do only specific composites arrive?
$endgroup$
– Shamim Akhtar
10 hours ago
|
show 2 more comments
$begingroup$
Solve the following... $309|(20^n-13^n-7^n)$ in $mathbb{Z}^+$.
I invested lotof time to it and finally went to WolframAlpha for help by typing...
Solve $309k=20^n-13^n-7^n$ over the integers. It returned the following... Note that all the primes are generated here. Can someone please explain why? Thanks! EDIT: the formulas suggested in the answerare too complicated, but this is so simple!
elementary-number-theory
New contributor
$endgroup$
Solve the following... $309|(20^n-13^n-7^n)$ in $mathbb{Z}^+$.
I invested lotof time to it and finally went to WolframAlpha for help by typing...
Solve $309k=20^n-13^n-7^n$ over the integers. It returned the following... Note that all the primes are generated here. Can someone please explain why? Thanks! EDIT: the formulas suggested in the answerare too complicated, but this is so simple!
elementary-number-theory
elementary-number-theory
New contributor
New contributor
edited 8 hours ago
Bill Dubuque
214k29198660
214k29198660
New contributor
asked 11 hours ago
Shamim AkhtarShamim Akhtar
469
469
New contributor
New contributor
$begingroup$
Is it false that it generates all the primes?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
It is easy to picture why there might be a solution for all prime $n$. However, the interesting part of the pattern is that $n=5cdot 5$ and $n=5cdot 7$ both appear, yet neither $n=3cdot 3$ nor $n=5cdot 5$ appear. Which odd composites admit solutions?
$endgroup$
– Mark Fischler
11 hours ago
$begingroup$
There are no even numbers ..
$endgroup$
– Shamim Akhtar
10 hours ago
3
$begingroup$
It seems like $n equiv pm1 pmod 6$.
$endgroup$
– md2perpe
10 hours ago
$begingroup$
Yeah right but why do only specific composites arrive?
$endgroup$
– Shamim Akhtar
10 hours ago
|
show 2 more comments
$begingroup$
Is it false that it generates all the primes?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
It is easy to picture why there might be a solution for all prime $n$. However, the interesting part of the pattern is that $n=5cdot 5$ and $n=5cdot 7$ both appear, yet neither $n=3cdot 3$ nor $n=5cdot 5$ appear. Which odd composites admit solutions?
$endgroup$
– Mark Fischler
11 hours ago
$begingroup$
There are no even numbers ..
$endgroup$
– Shamim Akhtar
10 hours ago
3
$begingroup$
It seems like $n equiv pm1 pmod 6$.
$endgroup$
– md2perpe
10 hours ago
$begingroup$
Yeah right but why do only specific composites arrive?
$endgroup$
– Shamim Akhtar
10 hours ago
$begingroup$
Is it false that it generates all the primes?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
Is it false that it generates all the primes?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
It is easy to picture why there might be a solution for all prime $n$. However, the interesting part of the pattern is that $n=5cdot 5$ and $n=5cdot 7$ both appear, yet neither $n=3cdot 3$ nor $n=5cdot 5$ appear. Which odd composites admit solutions?
$endgroup$
– Mark Fischler
11 hours ago
$begingroup$
It is easy to picture why there might be a solution for all prime $n$. However, the interesting part of the pattern is that $n=5cdot 5$ and $n=5cdot 7$ both appear, yet neither $n=3cdot 3$ nor $n=5cdot 5$ appear. Which odd composites admit solutions?
$endgroup$
– Mark Fischler
11 hours ago
$begingroup$
There are no even numbers ..
$endgroup$
– Shamim Akhtar
10 hours ago
$begingroup$
There are no even numbers ..
$endgroup$
– Shamim Akhtar
10 hours ago
3
3
$begingroup$
It seems like $n equiv pm1 pmod 6$.
$endgroup$
– md2perpe
10 hours ago
$begingroup$
It seems like $n equiv pm1 pmod 6$.
$endgroup$
– md2perpe
10 hours ago
$begingroup$
Yeah right but why do only specific composites arrive?
$endgroup$
– Shamim Akhtar
10 hours ago
$begingroup$
Yeah right but why do only specific composites arrive?
$endgroup$
– Shamim Akhtar
10 hours ago
|
show 2 more comments
3 Answers
3
active
oldest
votes
$begingroup$
$20^6 equiv 13^6 equiv 7^6 equiv 229 bmod 309$, so $20^n equiv 13^n + 7^n mod 309$ if and only if $20^{n+6} equiv 13^{n+6} + 7^{n+6} bmod 309$. Since it does work for $n=1$ and $n=5$, but not for $0$, $2$, $3$ or $4$, we find that $20^n equiv 13^n + 7^n bmod 309$ if and only if $n equiv 1$ or $5 bmod 6$.
All primes except $2$ and $3$ are congruent to $1$ or $5 bmod 6$. $2$ or $3$ can't divide a number congruent to $1$ or $5 bmod 6$, so it's not easy for a small number of this form to be composite. The first few composites are $25 = 5^2$, $35 = 5 cdot 7$, $49 = 7 cdot 7$, $55 = 5 cdot 11$, ...
$endgroup$
$begingroup$
+1 really cool.
$endgroup$
– Shamim Akhtar
10 hours ago
2
$begingroup$
@ShamimAkhtar If you had entered your sequence $,5,7,11,13,ldots, 65$ into OEIS you'd obtain a unique match: A007310 = Numbers congruent to $1$ or $5$ mod $6$. Once you know that the rest is easy. That's the first method one should try on problems like this (when the pattern is not already clear from prior knowledge).
$endgroup$
– Bill Dubuque
7 hours ago
1
$begingroup$
At mathoverflow.net/questions/328782/… there is a proof that in general $(a^2-ab+b^2)$ divides $a^{6k+r} -b^{6k+r} -(a-b)^{6k+r}$ for $r = 1, 5$. The question at hand is a particular case with $a=20, b=13$ where $(a^2-ab+b^2)=309$
$endgroup$
– Mark Fischler
5 hours ago
$begingroup$
@Mark We can also proceed analogously as I do here. (iirc this specific case is done in some other question here).
$endgroup$
– Bill Dubuque
4 hours ago
add a comment |
$begingroup$
So far you have only shown evidence that some of the primes are generated here, not all of them. It's interesting how many there are, though. There are some polynomials that generate an exorbitant number of primes. Here are some examples. You will likely have to do a lot more work than just exploring the idea numerically to prove that all primes are generated. Even if you do, these functions aren't super interesting because they produce non-primes as well. You won't know whether a number that is generated is prime or not until you test it for primality, so it's not like it can be used to generate the $n^{th}$ prime without having to do lots more computation. Quite cool, though.
$endgroup$
$begingroup$
Can you explain why this works?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
I tapped the show more but wolframalpha doesnt show more
$endgroup$
– Shamim Akhtar
11 hours ago
1
$begingroup$
I'm afraid I canot
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
Is this a standard question in mse? I too am afraid if any moderator marks this as off topic
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
It's a perfectly acceptable question, although it's probably much more difficult to answer than we think. As a result, you're probably either not going to get an answer or get a very strange one. I could be wrong, number theory isn't my thing.
$endgroup$
– Michael Stachowsky
10 hours ago
|
show 1 more comment
$begingroup$
A trivial observation is that it does not generate all the odd primes: $3$ is missing.
However, I can verify that all primes up to $59359$ (other than $2$ and $3$) are represented.
This has a lot do to with the fact that $(a^2-ab+^2)$ is a apparently factor of $a^n-b^n-(a-b)^n$ for all prime $n>3$. Here, $a=20, b=13$.
But that fact is at least as interesting as your observation, and I don't know how to prove it.
$endgroup$
$begingroup$
How can you verify?
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
Mathematica is very fast in determining residues mod 309.
$endgroup$
– Mark Fischler
10 hours ago
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$20^6 equiv 13^6 equiv 7^6 equiv 229 bmod 309$, so $20^n equiv 13^n + 7^n mod 309$ if and only if $20^{n+6} equiv 13^{n+6} + 7^{n+6} bmod 309$. Since it does work for $n=1$ and $n=5$, but not for $0$, $2$, $3$ or $4$, we find that $20^n equiv 13^n + 7^n bmod 309$ if and only if $n equiv 1$ or $5 bmod 6$.
All primes except $2$ and $3$ are congruent to $1$ or $5 bmod 6$. $2$ or $3$ can't divide a number congruent to $1$ or $5 bmod 6$, so it's not easy for a small number of this form to be composite. The first few composites are $25 = 5^2$, $35 = 5 cdot 7$, $49 = 7 cdot 7$, $55 = 5 cdot 11$, ...
$endgroup$
$begingroup$
+1 really cool.
$endgroup$
– Shamim Akhtar
10 hours ago
2
$begingroup$
@ShamimAkhtar If you had entered your sequence $,5,7,11,13,ldots, 65$ into OEIS you'd obtain a unique match: A007310 = Numbers congruent to $1$ or $5$ mod $6$. Once you know that the rest is easy. That's the first method one should try on problems like this (when the pattern is not already clear from prior knowledge).
$endgroup$
– Bill Dubuque
7 hours ago
1
$begingroup$
At mathoverflow.net/questions/328782/… there is a proof that in general $(a^2-ab+b^2)$ divides $a^{6k+r} -b^{6k+r} -(a-b)^{6k+r}$ for $r = 1, 5$. The question at hand is a particular case with $a=20, b=13$ where $(a^2-ab+b^2)=309$
$endgroup$
– Mark Fischler
5 hours ago
$begingroup$
@Mark We can also proceed analogously as I do here. (iirc this specific case is done in some other question here).
$endgroup$
– Bill Dubuque
4 hours ago
add a comment |
$begingroup$
$20^6 equiv 13^6 equiv 7^6 equiv 229 bmod 309$, so $20^n equiv 13^n + 7^n mod 309$ if and only if $20^{n+6} equiv 13^{n+6} + 7^{n+6} bmod 309$. Since it does work for $n=1$ and $n=5$, but not for $0$, $2$, $3$ or $4$, we find that $20^n equiv 13^n + 7^n bmod 309$ if and only if $n equiv 1$ or $5 bmod 6$.
All primes except $2$ and $3$ are congruent to $1$ or $5 bmod 6$. $2$ or $3$ can't divide a number congruent to $1$ or $5 bmod 6$, so it's not easy for a small number of this form to be composite. The first few composites are $25 = 5^2$, $35 = 5 cdot 7$, $49 = 7 cdot 7$, $55 = 5 cdot 11$, ...
$endgroup$
$begingroup$
+1 really cool.
$endgroup$
– Shamim Akhtar
10 hours ago
2
$begingroup$
@ShamimAkhtar If you had entered your sequence $,5,7,11,13,ldots, 65$ into OEIS you'd obtain a unique match: A007310 = Numbers congruent to $1$ or $5$ mod $6$. Once you know that the rest is easy. That's the first method one should try on problems like this (when the pattern is not already clear from prior knowledge).
$endgroup$
– Bill Dubuque
7 hours ago
1
$begingroup$
At mathoverflow.net/questions/328782/… there is a proof that in general $(a^2-ab+b^2)$ divides $a^{6k+r} -b^{6k+r} -(a-b)^{6k+r}$ for $r = 1, 5$. The question at hand is a particular case with $a=20, b=13$ where $(a^2-ab+b^2)=309$
$endgroup$
– Mark Fischler
5 hours ago
$begingroup$
@Mark We can also proceed analogously as I do here. (iirc this specific case is done in some other question here).
$endgroup$
– Bill Dubuque
4 hours ago
add a comment |
$begingroup$
$20^6 equiv 13^6 equiv 7^6 equiv 229 bmod 309$, so $20^n equiv 13^n + 7^n mod 309$ if and only if $20^{n+6} equiv 13^{n+6} + 7^{n+6} bmod 309$. Since it does work for $n=1$ and $n=5$, but not for $0$, $2$, $3$ or $4$, we find that $20^n equiv 13^n + 7^n bmod 309$ if and only if $n equiv 1$ or $5 bmod 6$.
All primes except $2$ and $3$ are congruent to $1$ or $5 bmod 6$. $2$ or $3$ can't divide a number congruent to $1$ or $5 bmod 6$, so it's not easy for a small number of this form to be composite. The first few composites are $25 = 5^2$, $35 = 5 cdot 7$, $49 = 7 cdot 7$, $55 = 5 cdot 11$, ...
$endgroup$
$20^6 equiv 13^6 equiv 7^6 equiv 229 bmod 309$, so $20^n equiv 13^n + 7^n mod 309$ if and only if $20^{n+6} equiv 13^{n+6} + 7^{n+6} bmod 309$. Since it does work for $n=1$ and $n=5$, but not for $0$, $2$, $3$ or $4$, we find that $20^n equiv 13^n + 7^n bmod 309$ if and only if $n equiv 1$ or $5 bmod 6$.
All primes except $2$ and $3$ are congruent to $1$ or $5 bmod 6$. $2$ or $3$ can't divide a number congruent to $1$ or $5 bmod 6$, so it's not easy for a small number of this form to be composite. The first few composites are $25 = 5^2$, $35 = 5 cdot 7$, $49 = 7 cdot 7$, $55 = 5 cdot 11$, ...
edited 10 hours ago
answered 10 hours ago
Robert IsraelRobert Israel
333k23223484
333k23223484
$begingroup$
+1 really cool.
$endgroup$
– Shamim Akhtar
10 hours ago
2
$begingroup$
@ShamimAkhtar If you had entered your sequence $,5,7,11,13,ldots, 65$ into OEIS you'd obtain a unique match: A007310 = Numbers congruent to $1$ or $5$ mod $6$. Once you know that the rest is easy. That's the first method one should try on problems like this (when the pattern is not already clear from prior knowledge).
$endgroup$
– Bill Dubuque
7 hours ago
1
$begingroup$
At mathoverflow.net/questions/328782/… there is a proof that in general $(a^2-ab+b^2)$ divides $a^{6k+r} -b^{6k+r} -(a-b)^{6k+r}$ for $r = 1, 5$. The question at hand is a particular case with $a=20, b=13$ where $(a^2-ab+b^2)=309$
$endgroup$
– Mark Fischler
5 hours ago
$begingroup$
@Mark We can also proceed analogously as I do here. (iirc this specific case is done in some other question here).
$endgroup$
– Bill Dubuque
4 hours ago
add a comment |
$begingroup$
+1 really cool.
$endgroup$
– Shamim Akhtar
10 hours ago
2
$begingroup$
@ShamimAkhtar If you had entered your sequence $,5,7,11,13,ldots, 65$ into OEIS you'd obtain a unique match: A007310 = Numbers congruent to $1$ or $5$ mod $6$. Once you know that the rest is easy. That's the first method one should try on problems like this (when the pattern is not already clear from prior knowledge).
$endgroup$
– Bill Dubuque
7 hours ago
1
$begingroup$
At mathoverflow.net/questions/328782/… there is a proof that in general $(a^2-ab+b^2)$ divides $a^{6k+r} -b^{6k+r} -(a-b)^{6k+r}$ for $r = 1, 5$. The question at hand is a particular case with $a=20, b=13$ where $(a^2-ab+b^2)=309$
$endgroup$
– Mark Fischler
5 hours ago
$begingroup$
@Mark We can also proceed analogously as I do here. (iirc this specific case is done in some other question here).
$endgroup$
– Bill Dubuque
4 hours ago
$begingroup$
+1 really cool.
$endgroup$
– Shamim Akhtar
10 hours ago
$begingroup$
+1 really cool.
$endgroup$
– Shamim Akhtar
10 hours ago
2
2
$begingroup$
@ShamimAkhtar If you had entered your sequence $,5,7,11,13,ldots, 65$ into OEIS you'd obtain a unique match: A007310 = Numbers congruent to $1$ or $5$ mod $6$. Once you know that the rest is easy. That's the first method one should try on problems like this (when the pattern is not already clear from prior knowledge).
$endgroup$
– Bill Dubuque
7 hours ago
$begingroup$
@ShamimAkhtar If you had entered your sequence $,5,7,11,13,ldots, 65$ into OEIS you'd obtain a unique match: A007310 = Numbers congruent to $1$ or $5$ mod $6$. Once you know that the rest is easy. That's the first method one should try on problems like this (when the pattern is not already clear from prior knowledge).
$endgroup$
– Bill Dubuque
7 hours ago
1
1
$begingroup$
At mathoverflow.net/questions/328782/… there is a proof that in general $(a^2-ab+b^2)$ divides $a^{6k+r} -b^{6k+r} -(a-b)^{6k+r}$ for $r = 1, 5$. The question at hand is a particular case with $a=20, b=13$ where $(a^2-ab+b^2)=309$
$endgroup$
– Mark Fischler
5 hours ago
$begingroup$
At mathoverflow.net/questions/328782/… there is a proof that in general $(a^2-ab+b^2)$ divides $a^{6k+r} -b^{6k+r} -(a-b)^{6k+r}$ for $r = 1, 5$. The question at hand is a particular case with $a=20, b=13$ where $(a^2-ab+b^2)=309$
$endgroup$
– Mark Fischler
5 hours ago
$begingroup$
@Mark We can also proceed analogously as I do here. (iirc this specific case is done in some other question here).
$endgroup$
– Bill Dubuque
4 hours ago
$begingroup$
@Mark We can also proceed analogously as I do here. (iirc this specific case is done in some other question here).
$endgroup$
– Bill Dubuque
4 hours ago
add a comment |
$begingroup$
So far you have only shown evidence that some of the primes are generated here, not all of them. It's interesting how many there are, though. There are some polynomials that generate an exorbitant number of primes. Here are some examples. You will likely have to do a lot more work than just exploring the idea numerically to prove that all primes are generated. Even if you do, these functions aren't super interesting because they produce non-primes as well. You won't know whether a number that is generated is prime or not until you test it for primality, so it's not like it can be used to generate the $n^{th}$ prime without having to do lots more computation. Quite cool, though.
$endgroup$
$begingroup$
Can you explain why this works?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
I tapped the show more but wolframalpha doesnt show more
$endgroup$
– Shamim Akhtar
11 hours ago
1
$begingroup$
I'm afraid I canot
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
Is this a standard question in mse? I too am afraid if any moderator marks this as off topic
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
It's a perfectly acceptable question, although it's probably much more difficult to answer than we think. As a result, you're probably either not going to get an answer or get a very strange one. I could be wrong, number theory isn't my thing.
$endgroup$
– Michael Stachowsky
10 hours ago
|
show 1 more comment
$begingroup$
So far you have only shown evidence that some of the primes are generated here, not all of them. It's interesting how many there are, though. There are some polynomials that generate an exorbitant number of primes. Here are some examples. You will likely have to do a lot more work than just exploring the idea numerically to prove that all primes are generated. Even if you do, these functions aren't super interesting because they produce non-primes as well. You won't know whether a number that is generated is prime or not until you test it for primality, so it's not like it can be used to generate the $n^{th}$ prime without having to do lots more computation. Quite cool, though.
$endgroup$
$begingroup$
Can you explain why this works?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
I tapped the show more but wolframalpha doesnt show more
$endgroup$
– Shamim Akhtar
11 hours ago
1
$begingroup$
I'm afraid I canot
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
Is this a standard question in mse? I too am afraid if any moderator marks this as off topic
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– Shamim Akhtar
10 hours ago
1
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It's a perfectly acceptable question, although it's probably much more difficult to answer than we think. As a result, you're probably either not going to get an answer or get a very strange one. I could be wrong, number theory isn't my thing.
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– Michael Stachowsky
10 hours ago
|
show 1 more comment
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So far you have only shown evidence that some of the primes are generated here, not all of them. It's interesting how many there are, though. There are some polynomials that generate an exorbitant number of primes. Here are some examples. You will likely have to do a lot more work than just exploring the idea numerically to prove that all primes are generated. Even if you do, these functions aren't super interesting because they produce non-primes as well. You won't know whether a number that is generated is prime or not until you test it for primality, so it's not like it can be used to generate the $n^{th}$ prime without having to do lots more computation. Quite cool, though.
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So far you have only shown evidence that some of the primes are generated here, not all of them. It's interesting how many there are, though. There are some polynomials that generate an exorbitant number of primes. Here are some examples. You will likely have to do a lot more work than just exploring the idea numerically to prove that all primes are generated. Even if you do, these functions aren't super interesting because they produce non-primes as well. You won't know whether a number that is generated is prime or not until you test it for primality, so it's not like it can be used to generate the $n^{th}$ prime without having to do lots more computation. Quite cool, though.
answered 11 hours ago
Michael StachowskyMichael Stachowsky
1,409418
1,409418
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Can you explain why this works?
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– Shamim Akhtar
11 hours ago
$begingroup$
I tapped the show more but wolframalpha doesnt show more
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– Shamim Akhtar
11 hours ago
1
$begingroup$
I'm afraid I canot
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
Is this a standard question in mse? I too am afraid if any moderator marks this as off topic
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
It's a perfectly acceptable question, although it's probably much more difficult to answer than we think. As a result, you're probably either not going to get an answer or get a very strange one. I could be wrong, number theory isn't my thing.
$endgroup$
– Michael Stachowsky
10 hours ago
|
show 1 more comment
$begingroup$
Can you explain why this works?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
I tapped the show more but wolframalpha doesnt show more
$endgroup$
– Shamim Akhtar
11 hours ago
1
$begingroup$
I'm afraid I canot
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
Is this a standard question in mse? I too am afraid if any moderator marks this as off topic
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
It's a perfectly acceptable question, although it's probably much more difficult to answer than we think. As a result, you're probably either not going to get an answer or get a very strange one. I could be wrong, number theory isn't my thing.
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
Can you explain why this works?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
Can you explain why this works?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
I tapped the show more but wolframalpha doesnt show more
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
I tapped the show more but wolframalpha doesnt show more
$endgroup$
– Shamim Akhtar
11 hours ago
1
1
$begingroup$
I'm afraid I canot
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
I'm afraid I canot
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
Is this a standard question in mse? I too am afraid if any moderator marks this as off topic
$endgroup$
– Shamim Akhtar
10 hours ago
$begingroup$
Is this a standard question in mse? I too am afraid if any moderator marks this as off topic
$endgroup$
– Shamim Akhtar
10 hours ago
1
1
$begingroup$
It's a perfectly acceptable question, although it's probably much more difficult to answer than we think. As a result, you're probably either not going to get an answer or get a very strange one. I could be wrong, number theory isn't my thing.
$endgroup$
– Michael Stachowsky
10 hours ago
$begingroup$
It's a perfectly acceptable question, although it's probably much more difficult to answer than we think. As a result, you're probably either not going to get an answer or get a very strange one. I could be wrong, number theory isn't my thing.
$endgroup$
– Michael Stachowsky
10 hours ago
|
show 1 more comment
$begingroup$
A trivial observation is that it does not generate all the odd primes: $3$ is missing.
However, I can verify that all primes up to $59359$ (other than $2$ and $3$) are represented.
This has a lot do to with the fact that $(a^2-ab+^2)$ is a apparently factor of $a^n-b^n-(a-b)^n$ for all prime $n>3$. Here, $a=20, b=13$.
But that fact is at least as interesting as your observation, and I don't know how to prove it.
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$begingroup$
How can you verify?
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– Shamim Akhtar
10 hours ago
1
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Mathematica is very fast in determining residues mod 309.
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– Mark Fischler
10 hours ago
add a comment |
$begingroup$
A trivial observation is that it does not generate all the odd primes: $3$ is missing.
However, I can verify that all primes up to $59359$ (other than $2$ and $3$) are represented.
This has a lot do to with the fact that $(a^2-ab+^2)$ is a apparently factor of $a^n-b^n-(a-b)^n$ for all prime $n>3$. Here, $a=20, b=13$.
But that fact is at least as interesting as your observation, and I don't know how to prove it.
$endgroup$
$begingroup$
How can you verify?
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
Mathematica is very fast in determining residues mod 309.
$endgroup$
– Mark Fischler
10 hours ago
add a comment |
$begingroup$
A trivial observation is that it does not generate all the odd primes: $3$ is missing.
However, I can verify that all primes up to $59359$ (other than $2$ and $3$) are represented.
This has a lot do to with the fact that $(a^2-ab+^2)$ is a apparently factor of $a^n-b^n-(a-b)^n$ for all prime $n>3$. Here, $a=20, b=13$.
But that fact is at least as interesting as your observation, and I don't know how to prove it.
$endgroup$
A trivial observation is that it does not generate all the odd primes: $3$ is missing.
However, I can verify that all primes up to $59359$ (other than $2$ and $3$) are represented.
This has a lot do to with the fact that $(a^2-ab+^2)$ is a apparently factor of $a^n-b^n-(a-b)^n$ for all prime $n>3$. Here, $a=20, b=13$.
But that fact is at least as interesting as your observation, and I don't know how to prove it.
edited 10 hours ago
answered 10 hours ago
Mark FischlerMark Fischler
34.6k12552
34.6k12552
$begingroup$
How can you verify?
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
Mathematica is very fast in determining residues mod 309.
$endgroup$
– Mark Fischler
10 hours ago
add a comment |
$begingroup$
How can you verify?
$endgroup$
– Shamim Akhtar
10 hours ago
1
$begingroup$
Mathematica is very fast in determining residues mod 309.
$endgroup$
– Mark Fischler
10 hours ago
$begingroup$
How can you verify?
$endgroup$
– Shamim Akhtar
10 hours ago
$begingroup$
How can you verify?
$endgroup$
– Shamim Akhtar
10 hours ago
1
1
$begingroup$
Mathematica is very fast in determining residues mod 309.
$endgroup$
– Mark Fischler
10 hours ago
$begingroup$
Mathematica is very fast in determining residues mod 309.
$endgroup$
– Mark Fischler
10 hours ago
add a comment |
Shamim Akhtar is a new contributor. Be nice, and check out our Code of Conduct.
Shamim Akhtar is a new contributor. Be nice, and check out our Code of Conduct.
Shamim Akhtar is a new contributor. Be nice, and check out our Code of Conduct.
Shamim Akhtar is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
Is it false that it generates all the primes?
$endgroup$
– Shamim Akhtar
11 hours ago
$begingroup$
It is easy to picture why there might be a solution for all prime $n$. However, the interesting part of the pattern is that $n=5cdot 5$ and $n=5cdot 7$ both appear, yet neither $n=3cdot 3$ nor $n=5cdot 5$ appear. Which odd composites admit solutions?
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– Mark Fischler
11 hours ago
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There are no even numbers ..
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– Shamim Akhtar
10 hours ago
3
$begingroup$
It seems like $n equiv pm1 pmod 6$.
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– md2perpe
10 hours ago
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Yeah right but why do only specific composites arrive?
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– Shamim Akhtar
10 hours ago