Distribution of prime numbers modulo $4$ Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Prime spiral distribution into quadrantsRandomness in prime numbersDistribution of prime numbers. Can one find all prime numbers?Induction on prime numbersWhy does Euclid write “Prime numbers are more than any assigned multitude of prime numbers.”A question about Prime Numbers (and its relation to RSA Asymmetric Cryptography)prime and irreducible elements $equiv 1$ modulo $4$Number of $k$th Roots modulo a prime?Modulo world of four remainder 1Density distribution of prime numbers modulo 13?
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Distribution of prime numbers modulo $4$
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Prime spiral distribution into quadrantsRandomness in prime numbersDistribution of prime numbers. Can one find all prime numbers?Induction on prime numbersWhy does Euclid write “Prime numbers are more than any assigned multitude of prime numbers.”A question about Prime Numbers (and its relation to RSA Asymmetric Cryptography)prime and irreducible elements $equiv 1$ modulo $4$Number of $k$th Roots modulo a prime?Modulo world of four remainder 1Density distribution of prime numbers modulo 13?
$begingroup$
Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction?
That is my specific question, but I would be interested to know if there exists a trend more generally, say for modulo any even.
number-theory prime-numbers
$endgroup$
add a comment |
$begingroup$
Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction?
That is my specific question, but I would be interested to know if there exists a trend more generally, say for modulo any even.
number-theory prime-numbers
$endgroup$
add a comment |
$begingroup$
Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction?
That is my specific question, but I would be interested to know if there exists a trend more generally, say for modulo any even.
number-theory prime-numbers
$endgroup$
Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction?
That is my specific question, but I would be interested to know if there exists a trend more generally, say for modulo any even.
number-theory prime-numbers
number-theory prime-numbers
edited 7 mins ago
YuiTo Cheng
2,65641037
2,65641037
asked 33 mins ago
Will SeathWill Seath
414
414
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.
$endgroup$
1
$begingroup$
I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $frac1phi(d)$ in each class.
$endgroup$
– Μάρκος Καραμέρης
27 mins ago
1
$begingroup$
Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue.
$endgroup$
– Peter
25 mins ago
$begingroup$
Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens?
$endgroup$
– Μάρκος Καραμέρης
21 mins ago
$begingroup$
I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often.
$endgroup$
– Peter
18 mins ago
add a comment |
$begingroup$
In general, if $gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $dfracpi(x)varphi(a)$ where $pi(x)$ is the number of primes $le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
$endgroup$
1
$begingroup$
(for $b$ coprime to $a$, of course)
$endgroup$
– Robert Israel
22 mins ago
$begingroup$
Yes of course ... updated :)
$endgroup$
– Nilotpal Kanti Sinha
21 mins ago
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.
$endgroup$
1
$begingroup$
I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $frac1phi(d)$ in each class.
$endgroup$
– Μάρκος Καραμέρης
27 mins ago
1
$begingroup$
Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue.
$endgroup$
– Peter
25 mins ago
$begingroup$
Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens?
$endgroup$
– Μάρκος Καραμέρης
21 mins ago
$begingroup$
I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often.
$endgroup$
– Peter
18 mins ago
add a comment |
$begingroup$
Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.
$endgroup$
1
$begingroup$
I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $frac1phi(d)$ in each class.
$endgroup$
– Μάρκος Καραμέρης
27 mins ago
1
$begingroup$
Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue.
$endgroup$
– Peter
25 mins ago
$begingroup$
Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens?
$endgroup$
– Μάρκος Καραμέρης
21 mins ago
$begingroup$
I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often.
$endgroup$
– Peter
18 mins ago
add a comment |
$begingroup$
Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.
$endgroup$
Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.
edited 17 mins ago
answered 31 mins ago
PeterPeter
49.3k1240138
49.3k1240138
1
$begingroup$
I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $frac1phi(d)$ in each class.
$endgroup$
– Μάρκος Καραμέρης
27 mins ago
1
$begingroup$
Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue.
$endgroup$
– Peter
25 mins ago
$begingroup$
Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens?
$endgroup$
– Μάρκος Καραμέρης
21 mins ago
$begingroup$
I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often.
$endgroup$
– Peter
18 mins ago
add a comment |
1
$begingroup$
I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $frac1phi(d)$ in each class.
$endgroup$
– Μάρκος Καραμέρης
27 mins ago
1
$begingroup$
Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue.
$endgroup$
– Peter
25 mins ago
$begingroup$
Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens?
$endgroup$
– Μάρκος Καραμέρης
21 mins ago
$begingroup$
I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often.
$endgroup$
– Peter
18 mins ago
1
1
$begingroup$
I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $frac1phi(d)$ in each class.
$endgroup$
– Μάρκος Καραμέρης
27 mins ago
$begingroup$
I would also note for future reference that the proof is given as an immediate consequence of Dirichlet’s Theorem and in general the primes are equidistributed modulo every $d$ with approximately $frac1phi(d)$ in each class.
$endgroup$
– Μάρκος Καραμέρης
27 mins ago
1
1
$begingroup$
Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue.
$endgroup$
– Peter
25 mins ago
$begingroup$
Generalization of Chebyshev's biase : The primes with a quadratic non-residue usually occur more often in practice than the primes with a quadratic residue.
$endgroup$
– Peter
25 mins ago
$begingroup$
Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens?
$endgroup$
– Μάρκος Καραμέρης
21 mins ago
$begingroup$
Interesting I haven't heard of that generalization before, is there any reason other than empirical evidence that this happens?
$endgroup$
– Μάρκος Καραμέρης
21 mins ago
$begingroup$
I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often.
$endgroup$
– Peter
18 mins ago
$begingroup$
I only know that Chebyshev noticed this phenomenon, no idea whether it has a mathematical reason. Also interesting : In the "race" between primes of the form $4k+1$ and $4k+3$ it is known that the lead switches infinite many often.
$endgroup$
– Peter
18 mins ago
add a comment |
$begingroup$
In general, if $gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $dfracpi(x)varphi(a)$ where $pi(x)$ is the number of primes $le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
$endgroup$
1
$begingroup$
(for $b$ coprime to $a$, of course)
$endgroup$
– Robert Israel
22 mins ago
$begingroup$
Yes of course ... updated :)
$endgroup$
– Nilotpal Kanti Sinha
21 mins ago
add a comment |
$begingroup$
In general, if $gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $dfracpi(x)varphi(a)$ where $pi(x)$ is the number of primes $le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
$endgroup$
1
$begingroup$
(for $b$ coprime to $a$, of course)
$endgroup$
– Robert Israel
22 mins ago
$begingroup$
Yes of course ... updated :)
$endgroup$
– Nilotpal Kanti Sinha
21 mins ago
add a comment |
$begingroup$
In general, if $gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $dfracpi(x)varphi(a)$ where $pi(x)$ is the number of primes $le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
$endgroup$
In general, if $gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $dfracpi(x)varphi(a)$ where $pi(x)$ is the number of primes $le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
edited 14 mins ago
Peter
49.3k1240138
49.3k1240138
answered 24 mins ago
Nilotpal Kanti SinhaNilotpal Kanti Sinha
4,73821641
4,73821641
1
$begingroup$
(for $b$ coprime to $a$, of course)
$endgroup$
– Robert Israel
22 mins ago
$begingroup$
Yes of course ... updated :)
$endgroup$
– Nilotpal Kanti Sinha
21 mins ago
add a comment |
1
$begingroup$
(for $b$ coprime to $a$, of course)
$endgroup$
– Robert Israel
22 mins ago
$begingroup$
Yes of course ... updated :)
$endgroup$
– Nilotpal Kanti Sinha
21 mins ago
1
1
$begingroup$
(for $b$ coprime to $a$, of course)
$endgroup$
– Robert Israel
22 mins ago
$begingroup$
(for $b$ coprime to $a$, of course)
$endgroup$
– Robert Israel
22 mins ago
$begingroup$
Yes of course ... updated :)
$endgroup$
– Nilotpal Kanti Sinha
21 mins ago
$begingroup$
Yes of course ... updated :)
$endgroup$
– Nilotpal Kanti Sinha
21 mins ago
add a comment |
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