A question about a certain type of primesIs there a prime of the given form?Primes and proofsHow many primes does Euclid's proof account for?$2n^2-lfloor m^brfloor=k$ has only finitely many integer solutionsAre they twin primes?Does there exist a positive irrational number $alpha $, such that for any positive integer $n$ the number $lfloor nalpha rfloor$ is not a prime?Is anything know about primes based on $2^n! pmod n!$ (I call them Prolonged Primes)?Enumerating arithmetic progressions of primesPiatetski-Shapiro PrimesQuestion about primes of the form $n^2+1$Root of the sum of two squared twin primes multiplied by $2*pi$
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A question about a certain type of primes
Is there a prime of the given form?Primes and proofsHow many primes does Euclid's proof account for?$2n^2-lfloor m^brfloor=k$ has only finitely many integer solutionsAre they twin primes?Does there exist a positive irrational number $alpha $, such that for any positive integer $n$ the number $lfloor nalpha rfloor$ is not a prime?Is anything know about primes based on $2^n! pmod n!$ (I call them Prolonged Primes)?Enumerating arithmetic progressions of primesPiatetski-Shapiro PrimesQuestion about primes of the form $n^2+1$Root of the sum of two squared twin primes multiplied by $2*pi$
$begingroup$
This is a computational problem, I don't know how much theoretical.
With Pari, just for fun, I found:
beginarrayc
2 & 19\
hline
3 & 29\
hline
5 & 487\
hline
7 &22053404036884180513958627370176317218668443714432289646932414656004065291773402535727759 \
hline
11 & 33223
endarray
The first column is the $n$-th prime $p_n$, whereas the second column is the prime $lfloor p_n*pi^krfloor=A(n)$ with the smallest possible $k$ positive integer, i mean k is the smallest positive integer for which $A(n)$ is prime. My question is: can exist two or more primes $A(n)$'s which are the same?
number-theory
asked Mar 22 at 15:26
homunculushomunculus
17910
$endgroup$
|
show 2 more comments
$begingroup$
This is a computational problem, I don't know how much theoretical.
With Pari, just for fun, I found:
beginarrayc
2 & 19\
hline
3 & 29\
hline
5 & 487\
hline
7 &22053404036884180513958627370176317218668443714432289646932414656004065291773402535727759 \
hline
11 & 33223
endarray
The first column is the $n$-th prime $p_n$, whereas the second column is the prime $lfloor p_n*pi^krfloor=A(n)$ with the smallest possible $k$ positive integer, i mean k is the smallest positive integer for which $A(n)$ is prime. My question is: can exist two or more primes $A(n)$'s which are the same?
number-theory
asked Mar 22 at 15:26
homunculushomunculus
17910
$endgroup$
$begingroup$
@Brian I mean the smallest possible k for which A(n) is prime
$endgroup$
– homunculus
Mar 22 at 15:35
$begingroup$
Is the existence of k obvious?
$endgroup$
– Lior B-S
Mar 22 at 15:37
$begingroup$
@Lior B-S no surely not obvious, but my question is if there are two or more A(n)'s which are the same.
$endgroup$
– homunculus
Mar 22 at 15:39
$begingroup$
These type of questions tend to be very difficult. How far did you check it on a computer? Maybe you will get lucky.
$endgroup$
– Lior B-S
Mar 22 at 15:40
1
$begingroup$
@Lior B-S from nothing...
$endgroup$
– homunculus
Mar 22 at 15:45
|
show 2 more comments
$begingroup$
This is a computational problem, I don't know how much theoretical.
With Pari, just for fun, I found:
beginarrayc
2 & 19\
hline
3 & 29\
hline
5 & 487\
hline
7 &22053404036884180513958627370176317218668443714432289646932414656004065291773402535727759 \
hline
11 & 33223
endarray
The first column is the $n$-th prime $p_n$, whereas the second column is the prime $lfloor p_n*pi^krfloor=A(n)$ with the smallest possible $k$ positive integer, i mean k is the smallest positive integer for which $A(n)$ is prime. My question is: can exist two or more primes $A(n)$'s which are the same?
number-theory
asked Mar 22 at 15:26
homunculushomunculus
17910
$endgroup$
This is a computational problem, I don't know how much theoretical.
With Pari, just for fun, I found:
beginarrayc
2 & 19\
hline
3 & 29\
hline
5 & 487\
hline
7 &22053404036884180513958627370176317218668443714432289646932414656004065291773402535727759 \
hline
11 & 33223
endarray
The first column is the $n$-th prime $p_n$, whereas the second column is the prime $lfloor p_n*pi^krfloor=A(n)$ with the smallest possible $k$ positive integer, i mean k is the smallest positive integer for which $A(n)$ is prime. My question is: can exist two or more primes $A(n)$'s which are the same?
number-theory
number-theory
asked Mar 22 at 15:26
homunculushomunculus
17910
asked Mar 22 at 15:26
homunculushomunculus
17910
edited Mar 22 at 15:36
homunculus
asked Mar 22 at 15:26
homunculushomunculus
17910
asked Mar 22 at 15:26
homunculushomunculus
17910
asked Mar 22 at 15:26
homunculushomunculus
17910
17910
$begingroup$
@Brian I mean the smallest possible k for which A(n) is prime
$endgroup$
– homunculus
Mar 22 at 15:35
$begingroup$
Is the existence of k obvious?
$endgroup$
– Lior B-S
Mar 22 at 15:37
$begingroup$
@Lior B-S no surely not obvious, but my question is if there are two or more A(n)'s which are the same.
$endgroup$
– homunculus
Mar 22 at 15:39
$begingroup$
These type of questions tend to be very difficult. How far did you check it on a computer? Maybe you will get lucky.
$endgroup$
– Lior B-S
Mar 22 at 15:40
1
$begingroup$
@Lior B-S from nothing...
$endgroup$
– homunculus
Mar 22 at 15:45
|
show 2 more comments
$begingroup$
@Brian I mean the smallest possible k for which A(n) is prime
$endgroup$
– homunculus
Mar 22 at 15:35
$begingroup$
Is the existence of k obvious?
$endgroup$
– Lior B-S
Mar 22 at 15:37
$begingroup$
@Lior B-S no surely not obvious, but my question is if there are two or more A(n)'s which are the same.
$endgroup$
– homunculus
Mar 22 at 15:39
$begingroup$
These type of questions tend to be very difficult. How far did you check it on a computer? Maybe you will get lucky.
$endgroup$
– Lior B-S
Mar 22 at 15:40
1
$begingroup$
@Lior B-S from nothing...
$endgroup$
– homunculus
Mar 22 at 15:45
$begingroup$
@Brian I mean the smallest possible k for which A(n) is prime
$endgroup$
– homunculus
Mar 22 at 15:35
$begingroup$
@Brian I mean the smallest possible k for which A(n) is prime
$endgroup$
– homunculus
Mar 22 at 15:35
$begingroup$
Is the existence of k obvious?
$endgroup$
– Lior B-S
Mar 22 at 15:37
$begingroup$
Is the existence of k obvious?
$endgroup$
– Lior B-S
Mar 22 at 15:37
$begingroup$
@Lior B-S no surely not obvious, but my question is if there are two or more A(n)'s which are the same.
$endgroup$
– homunculus
Mar 22 at 15:39
$begingroup$
@Lior B-S no surely not obvious, but my question is if there are two or more A(n)'s which are the same.
$endgroup$
– homunculus
Mar 22 at 15:39
$begingroup$
These type of questions tend to be very difficult. How far did you check it on a computer? Maybe you will get lucky.
$endgroup$
– Lior B-S
Mar 22 at 15:40
$begingroup$
These type of questions tend to be very difficult. How far did you check it on a computer? Maybe you will get lucky.
$endgroup$
– Lior B-S
Mar 22 at 15:40
1
1
$begingroup$
@Lior B-S from nothing...
$endgroup$
– homunculus
Mar 22 at 15:45
$begingroup$
@Lior B-S from nothing...
$endgroup$
– homunculus
Mar 22 at 15:45
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
$p_1 = 991, p_2=9781$ will both return $A(p_i)=30727$ for $k_1=3$ and $k_2=1$
I used a very simple brute force algo, looking only for values of $k$ small enough.
Edit other examples
$$beginarrayc
A(p_i)&p_1&k_1&p_2&k_2\
hline
30727&9781&1&991&3\
200579&20323&2&6469&3\
129499&41221&1&13121&2\
138283&44017&1&14011&2\
170063&54133&1&17231&2\
182617&58129&1&18503&2\
593429&60127&2&19139&3\
684091&69313&2&22063&3\
237689&75659&1&24083&2\
252869&80491&1&25621&2\
293729&93497&1&29761&2\
endarray$$
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
$endgroup$
$begingroup$
any other solution?
$endgroup$
– homunculus
Mar 22 at 16:14
$begingroup$
9781/991 is infact a good approximation of pi^2
$endgroup$
– homunculus
Mar 22 at 16:17
$begingroup$
yes there are plenty, even without looking too hard. I'll edit my post
$endgroup$
– Thomas Lesgourgues
Mar 25 at 10:50
add a comment |
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1 Answer
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1 Answer
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votes
$begingroup$
$p_1 = 991, p_2=9781$ will both return $A(p_i)=30727$ for $k_1=3$ and $k_2=1$
I used a very simple brute force algo, looking only for values of $k$ small enough.
Edit other examples
$$beginarrayc
A(p_i)&p_1&k_1&p_2&k_2\
hline
30727&9781&1&991&3\
200579&20323&2&6469&3\
129499&41221&1&13121&2\
138283&44017&1&14011&2\
170063&54133&1&17231&2\
182617&58129&1&18503&2\
593429&60127&2&19139&3\
684091&69313&2&22063&3\
237689&75659&1&24083&2\
252869&80491&1&25621&2\
293729&93497&1&29761&2\
endarray$$
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
$endgroup$
$begingroup$
any other solution?
$endgroup$
– homunculus
Mar 22 at 16:14
$begingroup$
9781/991 is infact a good approximation of pi^2
$endgroup$
– homunculus
Mar 22 at 16:17
$begingroup$
yes there are plenty, even without looking too hard. I'll edit my post
$endgroup$
– Thomas Lesgourgues
Mar 25 at 10:50
add a comment |
$begingroup$
$p_1 = 991, p_2=9781$ will both return $A(p_i)=30727$ for $k_1=3$ and $k_2=1$
I used a very simple brute force algo, looking only for values of $k$ small enough.
Edit other examples
$$beginarrayc
A(p_i)&p_1&k_1&p_2&k_2\
hline
30727&9781&1&991&3\
200579&20323&2&6469&3\
129499&41221&1&13121&2\
138283&44017&1&14011&2\
170063&54133&1&17231&2\
182617&58129&1&18503&2\
593429&60127&2&19139&3\
684091&69313&2&22063&3\
237689&75659&1&24083&2\
252869&80491&1&25621&2\
293729&93497&1&29761&2\
endarray$$
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
$endgroup$
$begingroup$
any other solution?
$endgroup$
– homunculus
Mar 22 at 16:14
$begingroup$
9781/991 is infact a good approximation of pi^2
$endgroup$
– homunculus
Mar 22 at 16:17
$begingroup$
yes there are plenty, even without looking too hard. I'll edit my post
$endgroup$
– Thomas Lesgourgues
Mar 25 at 10:50
add a comment |
$begingroup$
$p_1 = 991, p_2=9781$ will both return $A(p_i)=30727$ for $k_1=3$ and $k_2=1$
I used a very simple brute force algo, looking only for values of $k$ small enough.
Edit other examples
$$beginarrayc
A(p_i)&p_1&k_1&p_2&k_2\
hline
30727&9781&1&991&3\
200579&20323&2&6469&3\
129499&41221&1&13121&2\
138283&44017&1&14011&2\
170063&54133&1&17231&2\
182617&58129&1&18503&2\
593429&60127&2&19139&3\
684091&69313&2&22063&3\
237689&75659&1&24083&2\
252869&80491&1&25621&2\
293729&93497&1&29761&2\
endarray$$
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
$endgroup$
$p_1 = 991, p_2=9781$ will both return $A(p_i)=30727$ for $k_1=3$ and $k_2=1$
I used a very simple brute force algo, looking only for values of $k$ small enough.
Edit other examples
$$beginarrayc
A(p_i)&p_1&k_1&p_2&k_2\
hline
30727&9781&1&991&3\
200579&20323&2&6469&3\
129499&41221&1&13121&2\
138283&44017&1&14011&2\
170063&54133&1&17231&2\
182617&58129&1&18503&2\
593429&60127&2&19139&3\
684091&69313&2&22063&3\
237689&75659&1&24083&2\
252869&80491&1&25621&2\
293729&93497&1&29761&2\
endarray$$
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
edited Mar 25 at 10:58
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
answered Mar 22 at 16:07
Thomas LesgourguesThomas Lesgourgues
1,333220
1,333220
$begingroup$
any other solution?
$endgroup$
– homunculus
Mar 22 at 16:14
$begingroup$
9781/991 is infact a good approximation of pi^2
$endgroup$
– homunculus
Mar 22 at 16:17
$begingroup$
yes there are plenty, even without looking too hard. I'll edit my post
$endgroup$
– Thomas Lesgourgues
Mar 25 at 10:50
add a comment |
$begingroup$
any other solution?
$endgroup$
– homunculus
Mar 22 at 16:14
$begingroup$
9781/991 is infact a good approximation of pi^2
$endgroup$
– homunculus
Mar 22 at 16:17
$begingroup$
yes there are plenty, even without looking too hard. I'll edit my post
$endgroup$
– Thomas Lesgourgues
Mar 25 at 10:50
$begingroup$
any other solution?
$endgroup$
– homunculus
Mar 22 at 16:14
$begingroup$
any other solution?
$endgroup$
– homunculus
Mar 22 at 16:14
$begingroup$
9781/991 is infact a good approximation of pi^2
$endgroup$
– homunculus
Mar 22 at 16:17
$begingroup$
9781/991 is infact a good approximation of pi^2
$endgroup$
– homunculus
Mar 22 at 16:17
$begingroup$
yes there are plenty, even without looking too hard. I'll edit my post
$endgroup$
– Thomas Lesgourgues
Mar 25 at 10:50
$begingroup$
yes there are plenty, even without looking too hard. I'll edit my post
$endgroup$
– Thomas Lesgourgues
Mar 25 at 10:50
add a comment |
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$begingroup$
@Brian I mean the smallest possible k for which A(n) is prime
$endgroup$
– homunculus
Mar 22 at 15:35
$begingroup$
Is the existence of k obvious?
$endgroup$
– Lior B-S
Mar 22 at 15:37
$begingroup$
@Lior B-S no surely not obvious, but my question is if there are two or more A(n)'s which are the same.
$endgroup$
– homunculus
Mar 22 at 15:39
$begingroup$
These type of questions tend to be very difficult. How far did you check it on a computer? Maybe you will get lucky.
$endgroup$
– Lior B-S
Mar 22 at 15:40
1
$begingroup$
@Lior B-S from nothing...
$endgroup$
– homunculus
Mar 22 at 15:45