Extension of Dirichlet's Arithmetic Progression Theorem











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Dirichlet's Arithmetic Progression Theorem states that:




Given $a, binmathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $kinmathbb{Z^+}.$




For any given $a$ and $b$ let $K_{a,b}={kmid a+kb text{ is prime}}$.



Also consider another Dirichlet-Valid AP $c+jd$. Restrict $j$ to $j_kin K_{a,b}$.




Is $c+j_k d$ prime an infinite number of times?







prime-numbers arithmetic-progression







share|cite|improve this question


















  • 2




    You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)cdots f_r(x)$ is not identically 0 on $mathbf Z/(p)$.
    – KConrad
    Dec 1 at 13:44






  • 1




    The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers".
    – KConrad
    Dec 1 at 13:48















up vote
2
down vote

favorite












Dirichlet's Arithmetic Progression Theorem states that:




Given $a, binmathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $kinmathbb{Z^+}.$




For any given $a$ and $b$ let $K_{a,b}={kmid a+kb text{ is prime}}$.



Also consider another Dirichlet-Valid AP $c+jd$. Restrict $j$ to $j_kin K_{a,b}$.




Is $c+j_k d$ prime an infinite number of times?







prime-numbers arithmetic-progression







share|cite|improve this question


















  • 2




    You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)cdots f_r(x)$ is not identically 0 on $mathbf Z/(p)$.
    – KConrad
    Dec 1 at 13:44






  • 1




    The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers".
    – KConrad
    Dec 1 at 13:48













up vote
2
down vote

favorite









up vote
2
down vote

favorite











Dirichlet's Arithmetic Progression Theorem states that:




Given $a, binmathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $kinmathbb{Z^+}.$




For any given $a$ and $b$ let $K_{a,b}={kmid a+kb text{ is prime}}$.



Also consider another Dirichlet-Valid AP $c+jd$. Restrict $j$ to $j_kin K_{a,b}$.




Is $c+j_k d$ prime an infinite number of times?







prime-numbers arithmetic-progression







share|cite|improve this question













Dirichlet's Arithmetic Progression Theorem states that:




Given $a, binmathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $kinmathbb{Z^+}.$




For any given $a$ and $b$ let $K_{a,b}={kmid a+kb text{ is prime}}$.



Also consider another Dirichlet-Valid AP $c+jd$. Restrict $j$ to $j_kin K_{a,b}$.




Is $c+j_k d$ prime an infinite number of times?







prime-numbers arithmetic-progression




prime-numbers arithmetic-progression






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 1 at 9:54









JonMark Perry

9583718




9583718








  • 2




    You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)cdots f_r(x)$ is not identically 0 on $mathbf Z/(p)$.
    – KConrad
    Dec 1 at 13:44






  • 1




    The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers".
    – KConrad
    Dec 1 at 13:48














  • 2




    You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)cdots f_r(x)$ is not identically 0 on $mathbf Z/(p)$.
    – KConrad
    Dec 1 at 13:44






  • 1




    The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers".
    – KConrad
    Dec 1 at 13:48








2




2




You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)cdots f_r(x)$ is not identically 0 on $mathbf Z/(p)$.
– KConrad
Dec 1 at 13:44




You are asking if $a+kb$ and $c+kd$ can be prime at the same time for infinitely many $k$ when $(a,b) = 1$ and $(c,d) = 1$. A simple counterexample is $k$ and $k+1$. You should look up Schinzel's Hypothesis H (qualitative conjecture) or the Bateman-Horn conjecture (quantitative conjecture) to see when a finite list of nonconstant polynomials $f_1(x), ldots, f_r(x)$ with integer coefficients is expected to take on prime values infinitely often at the same time. A key "nonobvious" condition is that for each prime $p$, the product $f_1(x)cdots f_r(x)$ is not identically 0 on $mathbf Z/(p)$.
– KConrad
Dec 1 at 13:44




1




1




The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers".
– KConrad
Dec 1 at 13:48




The special case of this conjecture when the $f_i(x)$ are all linear goes back to Dickson (1904). Look up Dickson's conjecture on Wikipedia. The title of Dickson's paper is similar to the title of your post: "A New Extension of Dirichlet's Theorem on Prime Numbers".
– KConrad
Dec 1 at 13:48










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Consider the arithmetic progressions $2+3mathbb N$ and $1+5mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. On the other hand, if $1+5k$ is prime, then $k$ should be even. So, for any $kinmathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.



The same contradiction could be attained on the arithmetic progresions $2+1cdot mathbb N$ and $3+1cdotmathbb N$.






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    accepted










    Consider the arithmetic progressions $2+3mathbb N$ and $1+5mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. On the other hand, if $1+5k$ is prime, then $k$ should be even. So, for any $kinmathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.



    The same contradiction could be attained on the arithmetic progresions $2+1cdot mathbb N$ and $3+1cdotmathbb N$.






    share|cite|improve this answer



























      up vote
      7
      down vote



      accepted










      Consider the arithmetic progressions $2+3mathbb N$ and $1+5mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. On the other hand, if $1+5k$ is prime, then $k$ should be even. So, for any $kinmathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.



      The same contradiction could be attained on the arithmetic progresions $2+1cdot mathbb N$ and $3+1cdotmathbb N$.






      share|cite|improve this answer

























        up vote
        7
        down vote



        accepted







        up vote
        7
        down vote



        accepted






        Consider the arithmetic progressions $2+3mathbb N$ and $1+5mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. On the other hand, if $1+5k$ is prime, then $k$ should be even. So, for any $kinmathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.



        The same contradiction could be attained on the arithmetic progresions $2+1cdot mathbb N$ and $3+1cdotmathbb N$.






        share|cite|improve this answer














        Consider the arithmetic progressions $2+3mathbb N$ and $1+5mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. On the other hand, if $1+5k$ is prime, then $k$ should be even. So, for any $kinmathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.



        The same contradiction could be attained on the arithmetic progresions $2+1cdot mathbb N$ and $3+1cdotmathbb N$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 1 at 13:26

























        answered Dec 1 at 10:16









        Taras Banakh

        15.6k13190




        15.6k13190






























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