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Distribution of prime numbers modulo $4$

2

1

$begingroup$

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

share|cite|improve this question

edited 25 mins ago

YuiTo Cheng

2,65641037

asked 52 mins ago

Will SeathWill Seath

414

$endgroup$

add a comment | 

2

1

$begingroup$

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

share|cite|improve this question

edited 25 mins ago

YuiTo Cheng

2,65641037

asked 52 mins ago

Will SeathWill Seath

414

$endgroup$

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

share|cite|improve this question

edited 25 mins ago

YuiTo Cheng

2,65641037

asked 52 mins ago

Will SeathWill Seath

414

$endgroup$

share|cite|improve this question

edited 25 mins ago

YuiTo Cheng

2,65641037

asked 52 mins ago

Will SeathWill Seath

414

share|cite|improve this question

edited 25 mins ago

YuiTo Cheng

2,65641037

asked 52 mins ago

Will SeathWill Seath

414

edited 25 mins ago

YuiTo Cheng

2,65641037

edited 25 mins ago

YuiTo Cheng

2,65641037

asked 52 mins ago

Will SeathWill Seath

414

asked 52 mins ago

Will SeathWill Seath

414

2 Answers
2

active

oldest

votes

4

$begingroup$

Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.

share|cite|improve this answer

edited 35 mins ago

answered 49 mins ago

PeterPeter

49.3k1240138

$endgroup$

• 
  
  
  

  
  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
  45 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
  43 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
  39 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
  36 mins ago
  

  
  
  
  

add a comment | 

2

$begingroup$

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.

share|cite|improve this answer

edited 32 mins ago

Peter

49.3k1240138

answered 42 mins ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

4,73821641

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  (for $b$ coprime to $a$, of course)
  $endgroup$
  – Robert Israel
  40 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes of course ... updated :)
  $endgroup$
  – Nilotpal Kanti Sinha
  40 mins ago
  
  

  
  
  
  

add a comment | 

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4

$begingroup$

Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.

share|cite|improve this answer

edited 35 mins ago

answered 49 mins ago

PeterPeter

49.3k1240138

$endgroup$

• 
  
  
  

  
  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
  45 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
  43 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
  39 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
  36 mins ago
  

  
  
  
  

add a comment | 

4

$begingroup$

Despite of Chebychev's bias (more primes with residue $3$ occur usually in practice), in the long run, the ratio is $1:1$.

share|cite|improve this answer

edited 35 mins ago

answered 49 mins ago

PeterPeter

49.3k1240138

$endgroup$

• 
  
  
  

  
  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
  45 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
  43 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
  39 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
  36 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 $1:1$.

share|cite|improve this answer

edited 35 mins ago

answered 49 mins ago

PeterPeter

49.3k1240138

$endgroup$

share|cite|improve this answer

edited 35 mins ago

answered 49 mins ago

PeterPeter

49.3k1240138

answered 49 mins ago

PeterPeter

49.3k1240138

answered 49 mins ago

PeterPeter

49.3k1240138

2

$begingroup$

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.

share|cite|improve this answer

edited 32 mins ago

Peter

49.3k1240138

answered 42 mins ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

4,73821641

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  (for $b$ coprime to $a$, of course)
  $endgroup$
  – Robert Israel
  40 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes of course ... updated :)
  $endgroup$
  – Nilotpal Kanti Sinha
  40 mins ago
  
  

  
  
  
  

add a comment | 

2

$begingroup$

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.

share|cite|improve this answer

edited 32 mins ago

Peter

49.3k1240138

answered 42 mins ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

4,73821641

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  (for $b$ coprime to $a$, of course)
  $endgroup$
  – Robert Israel
  40 mins ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes of course ... updated :)
  $endgroup$
  – Nilotpal Kanti Sinha
  40 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.

share|cite|improve this answer

edited 32 mins ago

Peter

49.3k1240138

answered 42 mins ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

4,73821641

$endgroup$

share|cite|improve this answer

edited 32 mins ago

Peter

49.3k1240138

answered 42 mins ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

4,73821641

edited 32 mins ago

Peter

49.3k1240138

edited 32 mins ago

Peter

49.3k1240138

answered 42 mins ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

4,73821641

answered 42 mins ago

Nilotpal Kanti SinhaNilotpal Kanti Sinha

4,73821641