Extension of Dirichlet's Arithmetic Progression Theorem
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
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
add a comment |
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
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
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 '18 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 '18 at 13:48
add a comment |
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
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
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
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
asked Dec 1 '18 at 9:54
JonMark Perry
9683718
9683718
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 '18 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 '18 at 13:48
add a comment |
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 '18 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 '18 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 '18 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 '18 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 '18 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 '18 at 13:48
add a comment |
1 Answer
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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 $3+1cdotmathbb N$.
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
add a comment |
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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 $3+1cdotmathbb N$.
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
add a comment |
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 $3+1cdotmathbb N$.
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
add a comment |
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 $3+1cdotmathbb N$.
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
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 $3+1cdotmathbb N$.
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
edited Dec 1 '18 at 13:26
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
answered Dec 1 '18 at 10:16
Taras Banakh
15.6k13190
15.6k13190
add a comment |
add a comment |
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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 '18 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 '18 at 13:48