$n$ such $n!+1$ and $n!-1$ are both primes
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I am looking for positive integers $n$ such that $n!+1$ and $n!-1$ are both primes.
Looking in OEIS Lists, I found that the only known $n$ such $n!+1$ and $n!-1$ are both primes is $n=3$ (giving $5$ and $7$).
Is it known if $3$ is only value for $n$ or is it still an open problem ?
number-theory elementary-number-theory prime-numbers prime-twins
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
asked Jan 13 '17 at 16:52
BenLaz
476215
|
show 4 more comments
up vote
4
down vote
favorite
I am looking for positive integers $n$ such that $n!+1$ and $n!-1$ are both primes.
Looking in OEIS Lists, I found that the only known $n$ such $n!+1$ and $n!-1$ are both primes is $n=3$ (giving $5$ and $7$).
Is it known if $3$ is only value for $n$ or is it still an open problem ?
number-theory elementary-number-theory prime-numbers prime-twins
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
asked Jan 13 '17 at 16:52
BenLaz
476215
It's a well-known conjecture of Erdős for the $n!+1$ case, actually...
– Parcly Taxel
Jan 13 '17 at 16:55
Take a look at this en.wikipedia.org/wiki/Factorial_prime
– kingW3
Jan 13 '17 at 16:58
@kingW3 : I took a look to this article, but they don't speak about the case of $n!+1$ and $n!-1$ being both prime
– BenLaz
Jan 13 '17 at 17:01
1
Did you read the line labeled "COMMENTS"?
– Nate
Jan 13 '17 at 17:13
2
I don't know any source, if you want I can sketch why the "standard" heuristics (i.e. a probabilistic model) suggests that we should only expect finitely many such numbers. Proving such heuristics actually hold is in general very difficult though and I'd expect this to be about as hard as say proving there are finitely many Fermat primes.
– Nate
Jan 13 '17 at 17:40
|
show 4 more comments
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I am looking for positive integers $n$ such that $n!+1$ and $n!-1$ are both primes.
Looking in OEIS Lists, I found that the only known $n$ such $n!+1$ and $n!-1$ are both primes is $n=3$ (giving $5$ and $7$).
Is it known if $3$ is only value for $n$ or is it still an open problem ?
number-theory elementary-number-theory prime-numbers prime-twins
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
asked Jan 13 '17 at 16:52
BenLaz
476215
I am looking for positive integers $n$ such that $n!+1$ and $n!-1$ are both primes.
Looking in OEIS Lists, I found that the only known $n$ such $n!+1$ and $n!-1$ are both primes is $n=3$ (giving $5$ and $7$).
Is it known if $3$ is only value for $n$ or is it still an open problem ?
number-theory elementary-number-theory prime-numbers prime-twins
number-theory elementary-number-theory prime-numbers prime-twins
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
asked Jan 13 '17 at 16:52
BenLaz
476215
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
asked Jan 13 '17 at 16:52
BenLaz
476215
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
edited Nov 21 at 11:50
José Carlos Santos
144k20113214
144k20113214
asked Jan 13 '17 at 16:52
BenLaz
476215
asked Jan 13 '17 at 16:52
BenLaz
476215
asked Jan 13 '17 at 16:52
BenLaz
476215
476215
It's a well-known conjecture of Erdős for the $n!+1$ case, actually...
– Parcly Taxel
Jan 13 '17 at 16:55
Take a look at this en.wikipedia.org/wiki/Factorial_prime
– kingW3
Jan 13 '17 at 16:58
@kingW3 : I took a look to this article, but they don't speak about the case of $n!+1$ and $n!-1$ being both prime
– BenLaz
Jan 13 '17 at 17:01
1
Did you read the line labeled "COMMENTS"?
– Nate
Jan 13 '17 at 17:13
2
I don't know any source, if you want I can sketch why the "standard" heuristics (i.e. a probabilistic model) suggests that we should only expect finitely many such numbers. Proving such heuristics actually hold is in general very difficult though and I'd expect this to be about as hard as say proving there are finitely many Fermat primes.
– Nate
Jan 13 '17 at 17:40
|
show 4 more comments
It's a well-known conjecture of Erdős for the $n!+1$ case, actually...
– Parcly Taxel
Jan 13 '17 at 16:55
Take a look at this en.wikipedia.org/wiki/Factorial_prime
– kingW3
Jan 13 '17 at 16:58
@kingW3 : I took a look to this article, but they don't speak about the case of $n!+1$ and $n!-1$ being both prime
– BenLaz
Jan 13 '17 at 17:01
1
Did you read the line labeled "COMMENTS"?
– Nate
Jan 13 '17 at 17:13
2
I don't know any source, if you want I can sketch why the "standard" heuristics (i.e. a probabilistic model) suggests that we should only expect finitely many such numbers. Proving such heuristics actually hold is in general very difficult though and I'd expect this to be about as hard as say proving there are finitely many Fermat primes.
– Nate
Jan 13 '17 at 17:40
It's a well-known conjecture of Erdős for the $n!+1$ case, actually...
– Parcly Taxel
Jan 13 '17 at 16:55
It's a well-known conjecture of Erdős for the $n!+1$ case, actually...
– Parcly Taxel
Jan 13 '17 at 16:55
Take a look at this en.wikipedia.org/wiki/Factorial_prime
– kingW3
Jan 13 '17 at 16:58
Take a look at this en.wikipedia.org/wiki/Factorial_prime
– kingW3
Jan 13 '17 at 16:58
@kingW3 : I took a look to this article, but they don't speak about the case of $n!+1$ and $n!-1$ being both prime
– BenLaz
Jan 13 '17 at 17:01
@kingW3 : I took a look to this article, but they don't speak about the case of $n!+1$ and $n!-1$ being both prime
– BenLaz
Jan 13 '17 at 17:01
1
1
Did you read the line labeled "COMMENTS"?
– Nate
Jan 13 '17 at 17:13
Did you read the line labeled "COMMENTS"?
– Nate
Jan 13 '17 at 17:13
2
2
I don't know any source, if you want I can sketch why the "standard" heuristics (i.e. a probabilistic model) suggests that we should only expect finitely many such numbers. Proving such heuristics actually hold is in general very difficult though and I'd expect this to be about as hard as say proving there are finitely many Fermat primes.
– Nate
Jan 13 '17 at 17:40
I don't know any source, if you want I can sketch why the "standard" heuristics (i.e. a probabilistic model) suggests that we should only expect finitely many such numbers. Proving such heuristics actually hold is in general very difficult though and I'd expect this to be about as hard as say proving there are finitely many Fermat primes.
– Nate
Jan 13 '17 at 17:40
|
show 4 more comments
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It's a well-known conjecture of Erdős for the $n!+1$ case, actually...
– Parcly Taxel
Jan 13 '17 at 16:55
Take a look at this en.wikipedia.org/wiki/Factorial_prime
– kingW3
Jan 13 '17 at 16:58
@kingW3 : I took a look to this article, but they don't speak about the case of $n!+1$ and $n!-1$ being both prime
– BenLaz
Jan 13 '17 at 17:01
1
Did you read the line labeled "COMMENTS"?
– Nate
Jan 13 '17 at 17:13
2
I don't know any source, if you want I can sketch why the "standard" heuristics (i.e. a probabilistic model) suggests that we should only expect finitely many such numbers. Proving such heuristics actually hold is in general very difficult though and I'd expect this to be about as hard as say proving there are finitely many Fermat primes.
– Nate
Jan 13 '17 at 17:40