Mixing
Find all $n$ such that the following is prime [closed]
1
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Find all positive integers $n$ for which $(1+n+n^2+...+n^n)^2-n^n$ is prime.
number-theory elementary-number-theory prime-numbers integers
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closed as off-topic by Servaes, Peter, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos Jan 24 at 13:25
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "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." – Servaes, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
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1
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Find all positive integers $n$ for which $(1+n+n^2+...+n^n)^2-n^n$ is prime.
number-theory elementary-number-theory prime-numbers integers
$endgroup$
closed as off-topic by Servaes, Peter, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos Jan 24 at 13:25
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "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." – Servaes, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
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Welcome to Math.SE. Take a look at How to ask a good question at Math.SE. To avoid downvotes and closing you should add your own efforts to the question, and tell us where you got stuck.
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– drhab
Jan 24 at 9:45
2
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One such $n$ is 3, which gives the prime 997. Conjecture: this is the only one? There are no other solutions with $nle 100,$ as I have checked with Mathematica.
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– Reiner Martin
Jan 24 at 9:45
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Actually, there is another solution for $n=215.$ No further solutions with $nle 1000.$
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– Reiner Martin
Jan 24 at 9:51
1
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Are you sure that 215 works though? It would be way easier to just show that 3 is the only one
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– user637829
Jan 24 at 10:03
1
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@JonSmith With the help of Wolfram alpha, I found the expression which can be proven to be always composite (See answer below). Concerning the prime for $n=215$ (with the wrong formula). Since the number passed $300$ random bases in the Miller-Rabin-test, the doubt of the primality was only theoretical.
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– Peter
Jan 24 at 10:50
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show 3 more comments
1
1
1
$begingroup$
Find all positive integers $n$ for which $(1+n+n^2+...+n^n)^2-n^n$ is prime.
number-theory elementary-number-theory prime-numbers integers
$endgroup$
Find all positive integers $n$ for which $(1+n+n^2+...+n^n)^2-n^n$ is prime.
number-theory elementary-number-theory prime-numbers integers
number-theory elementary-number-theory prime-numbers integers
asked Jan 24 at 9:43
user637829
closed as off-topic by Servaes, Peter, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos Jan 24 at 13:25
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "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." – Servaes, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Servaes, Peter, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos Jan 24 at 13:25
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "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." – Servaes, Jyrki Lahtonen, ΘΣΦGenSan, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Welcome to Math.SE. Take a look at How to ask a good question at Math.SE. To avoid downvotes and closing you should add your own efforts to the question, and tell us where you got stuck.
$endgroup$
– drhab
Jan 24 at 9:45
2
$begingroup$
One such $n$ is 3, which gives the prime 997. Conjecture: this is the only one? There are no other solutions with $nle 100,$ as I have checked with Mathematica.
$endgroup$
– Reiner Martin
Jan 24 at 9:45
$begingroup$
Actually, there is another solution for $n=215.$ No further solutions with $nle 1000.$
$endgroup$
– Reiner Martin
Jan 24 at 9:51
1
$begingroup$
Are you sure that 215 works though? It would be way easier to just show that 3 is the only one
$endgroup$
– user637829
Jan 24 at 10:03
1
$begingroup$
@JonSmith With the help of Wolfram alpha, I found the expression which can be proven to be always composite (See answer below). Concerning the prime for $n=215$ (with the wrong formula). Since the number passed $300$ random bases in the Miller-Rabin-test, the doubt of the primality was only theoretical.
$endgroup$
– Peter
Jan 24 at 10:50
|
show 3 more comments
$begingroup$
Welcome to Math.SE. Take a look at How to ask a good question at Math.SE. To avoid downvotes and closing you should add your own efforts to the question, and tell us where you got stuck.
$endgroup$
– drhab
Jan 24 at 9:45
2
$begingroup$
One such $n$ is 3, which gives the prime 997. Conjecture: this is the only one? There are no other solutions with $nle 100,$ as I have checked with Mathematica.
$endgroup$
– Reiner Martin
Jan 24 at 9:45
$begingroup$
Actually, there is another solution for $n=215.$ No further solutions with $nle 1000.$
$endgroup$
– Reiner Martin
Jan 24 at 9:51
1
$begingroup$
Are you sure that 215 works though? It would be way easier to just show that 3 is the only one
$endgroup$
– user637829
Jan 24 at 10:03
1
$begingroup$
@JonSmith With the help of Wolfram alpha, I found the expression which can be proven to be always composite (See answer below). Concerning the prime for $n=215$ (with the wrong formula). Since the number passed $300$ random bases in the Miller-Rabin-test, the doubt of the primality was only theoretical.
$endgroup$
– Peter
Jan 24 at 10:50
$begingroup$
Welcome to Math.SE. Take a look at How to ask a good question at Math.SE. To avoid downvotes and closing you should add your own efforts to the question, and tell us where you got stuck.
$endgroup$
– drhab
Jan 24 at 9:45
$begingroup$
Welcome to Math.SE. Take a look at How to ask a good question at Math.SE. To avoid downvotes and closing you should add your own efforts to the question, and tell us where you got stuck.
$endgroup$
– drhab
Jan 24 at 9:45
2
2
$begingroup$
One such $n$ is 3, which gives the prime 997. Conjecture: this is the only one? There are no other solutions with $nle 100,$ as I have checked with Mathematica.
$endgroup$
– Reiner Martin
Jan 24 at 9:45
$begingroup$
One such $n$ is 3, which gives the prime 997. Conjecture: this is the only one? There are no other solutions with $nle 100,$ as I have checked with Mathematica.
$endgroup$
– Reiner Martin
Jan 24 at 9:45
$begingroup$
Actually, there is another solution for $n=215.$ No further solutions with $nle 1000.$
$endgroup$
– Reiner Martin
Jan 24 at 9:51
$begingroup$
Actually, there is another solution for $n=215.$ No further solutions with $nle 1000.$
$endgroup$
– Reiner Martin
Jan 24 at 9:51
1
1
$begingroup$
Are you sure that 215 works though? It would be way easier to just show that 3 is the only one
$endgroup$
– user637829
Jan 24 at 10:03
$begingroup$
Are you sure that 215 works though? It would be way easier to just show that 3 is the only one
$endgroup$
– user637829
Jan 24 at 10:03
1
1
$begingroup$
@JonSmith With the help of Wolfram alpha, I found the expression which can be proven to be always composite (See answer below). Concerning the prime for $n=215$ (with the wrong formula). Since the number passed $300$ random bases in the Miller-Rabin-test, the doubt of the primality was only theoretical.
$endgroup$
– Peter
Jan 24 at 10:50
$begingroup$
@JonSmith With the help of Wolfram alpha, I found the expression which can be proven to be always composite (See answer below). Concerning the prime for $n=215$ (with the wrong formula). Since the number passed $300$ random bases in the Miller-Rabin-test, the doubt of the primality was only theoretical.
$endgroup$
– Peter
Jan 24 at 10:50
|
show 3 more comments
1 Answer
1
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oldest
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3
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Assume $n$ is a positive integer greater than $1$.
The sum $$1+n+n^2+cdots n^n$$ is a geometric series with value $$frac{n^{n+1}-1}{n-1}$$
Hence the number can also be expressed as $$frac{(n^n-1)(n^{n+2}-1)}{(n-1)^2}$$ which is composite for every integer $n>1$
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
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$begingroup$
Checks out (+1). Too bad the question is kinda dismal. Unmotivated for starters.
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– Jyrki Lahtonen
Jan 24 at 10:54
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
3
$begingroup$
Assume $n$ is a positive integer greater than $1$.
The sum $$1+n+n^2+cdots n^n$$ is a geometric series with value $$frac{n^{n+1}-1}{n-1}$$
Hence the number can also be expressed as $$frac{(n^n-1)(n^{n+2}-1)}{(n-1)^2}$$ which is composite for every integer $n>1$
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
$endgroup$
$begingroup$
Checks out (+1). Too bad the question is kinda dismal. Unmotivated for starters.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 10:54
add a comment |
3
$begingroup$
Assume $n$ is a positive integer greater than $1$.
The sum $$1+n+n^2+cdots n^n$$ is a geometric series with value $$frac{n^{n+1}-1}{n-1}$$
Hence the number can also be expressed as $$frac{(n^n-1)(n^{n+2}-1)}{(n-1)^2}$$ which is composite for every integer $n>1$
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
$endgroup$
$begingroup$
Checks out (+1). Too bad the question is kinda dismal. Unmotivated for starters.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 10:54
add a comment |
3
3
3
$begingroup$
Assume $n$ is a positive integer greater than $1$.
The sum $$1+n+n^2+cdots n^n$$ is a geometric series with value $$frac{n^{n+1}-1}{n-1}$$
Hence the number can also be expressed as $$frac{(n^n-1)(n^{n+2}-1)}{(n-1)^2}$$ which is composite for every integer $n>1$
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
$endgroup$
Assume $n$ is a positive integer greater than $1$.
The sum $$1+n+n^2+cdots n^n$$ is a geometric series with value $$frac{n^{n+1}-1}{n-1}$$
Hence the number can also be expressed as $$frac{(n^n-1)(n^{n+2}-1)}{(n-1)^2}$$ which is composite for every integer $n>1$
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
answered Jan 24 at 10:47
PeterPeter
48.8k1139136
48.8k1139136
$begingroup$
Checks out (+1). Too bad the question is kinda dismal. Unmotivated for starters.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 10:54
add a comment |
$begingroup$
Checks out (+1). Too bad the question is kinda dismal. Unmotivated for starters.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 10:54
$begingroup$
Checks out (+1). Too bad the question is kinda dismal. Unmotivated for starters.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 10:54
$begingroup$
Checks out (+1). Too bad the question is kinda dismal. Unmotivated for starters.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 10:54
add a comment |
$begingroup$
Welcome to Math.SE. Take a look at How to ask a good question at Math.SE. To avoid downvotes and closing you should add your own efforts to the question, and tell us where you got stuck.
$endgroup$
– drhab
Jan 24 at 9:45
2
$begingroup$
One such $n$ is 3, which gives the prime 997. Conjecture: this is the only one? There are no other solutions with $nle 100,$ as I have checked with Mathematica.
$endgroup$
– Reiner Martin
Jan 24 at 9:45
$begingroup$
Actually, there is another solution for $n=215.$ No further solutions with $nle 1000.$
$endgroup$
– Reiner Martin
Jan 24 at 9:51
1
$begingroup$
Are you sure that 215 works though? It would be way easier to just show that 3 is the only one
$endgroup$
– user637829
Jan 24 at 10:03
1
$begingroup$
@JonSmith With the help of Wolfram alpha, I found the expression which can be proven to be always composite (See answer below). Concerning the prime for $n=215$ (with the wrong formula). Since the number passed $300$ random bases in the Miller-Rabin-test, the doubt of the primality was only theoretical.
$endgroup$
– Peter
Jan 24 at 10:50