Distribution of prime numbers modulo $4$












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 3 mins ago









      YuiTo Cheng

      2,65641037




      2,65641037










      asked 30 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 $frac{1}{phi(d)}$ in each class.
            $endgroup$
            – Μάρκος Καραμέρης
            23 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
            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
            17 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
            14 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 $dfrac{pi(x)}{varphi(a)}$ 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
            18 mins ago










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














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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 $frac{1}{phi(d)}$ in each class.
            $endgroup$
            – Μάρκος Καραμέρης
            23 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
            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
            17 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
            14 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 $frac{1}{phi(d)}$ in each class.
            $endgroup$
            – Μάρκος Καραμέρης
            23 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
            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
            17 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
            14 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 13 mins ago

























          answered 27 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 $frac{1}{phi(d)}$ in each class.
            $endgroup$
            – Μάρκος Καραμέρης
            23 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
            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
            17 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
            14 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 $frac{1}{phi(d)}$ in each class.
            $endgroup$
            – Μάρκος Καραμέρης
            23 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
            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
            17 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
            14 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 $frac{1}{phi(d)}$ in each class.
          $endgroup$
          – Μάρκος Καραμέρης
          23 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 $frac{1}{phi(d)}$ in each class.
          $endgroup$
          – Μάρκος Καραμέρης
          23 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
          21 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
          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
          17 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
          17 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
          14 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
          14 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 $dfrac{pi(x)}{varphi(a)}$ 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
            18 mins ago










          • $begingroup$
            Yes of course ... updated :)
            $endgroup$
            – Nilotpal Kanti Sinha
            17 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 $dfrac{pi(x)}{varphi(a)}$ 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
            18 mins ago










          • $begingroup$
            Yes of course ... updated :)
            $endgroup$
            – Nilotpal Kanti Sinha
            17 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 $dfrac{pi(x)}{varphi(a)}$ 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 $dfrac{pi(x)}{varphi(a)}$ 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 10 mins ago









          Peter

          49.3k1240138




          49.3k1240138










          answered 20 mins ago









          Nilotpal Kanti SinhaNilotpal Kanti Sinha

          4,73821641




          4,73821641








          • 1




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










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
















          • 1




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










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










          1




          1




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




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












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






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




















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