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?










2












$begingroup$


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











$endgroup$
















    2












    $begingroup$


    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











    $endgroup$














      2












      2








      2





      $begingroup$


      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











      $endgroup$




      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 12 mins ago









      YuiTo Cheng

      2,65641037




      2,65641037










      asked 39 mins ago









      Will SeathWill Seath

      414




      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











          $endgroup$








          • 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
            32 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
            30 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
            26 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
            23 mins ago


















          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











          $endgroup$








          • 1




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










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












          Your Answer








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          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          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











          $endgroup$








          • 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
            32 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
            30 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
            26 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
            23 mins ago















          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











          $endgroup$








          • 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
            32 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
            30 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
            26 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
            23 mins ago













          4












          4








          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











          $endgroup$



          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














          share|cite|improve this answer



          share|cite|improve this answer








          edited 22 mins ago

























          answered 36 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
            32 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
            30 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
            26 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
            23 mins ago












          • 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
            32 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
            30 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
            26 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
            23 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
          32 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
          32 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
          30 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
          30 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
          26 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
          26 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
          23 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
          23 mins ago











          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











          $endgroup$








          • 1




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










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
















          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











          $endgroup$








          • 1




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










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














          2












          2








          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











          $endgroup$



          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.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 19 mins ago









          Peter

          49.3k1240138




          49.3k1240138










          answered 29 mins ago









          Nilotpal Kanti SinhaNilotpal Kanti Sinha

          4,73821641




          4,73821641







          • 1




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










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













          • 1




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










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








          1




          1




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




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












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





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


















          draft saved

          draft discarded
















































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