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 at 9:54
JonMark Perry
9583718
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 at 9:54
JonMark Perry
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
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 at 9:54
JonMark Perry
9583718
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 at 9:54
JonMark Perry
9583718
asked Dec 1 at 9:54
JonMark Perry
9583718
asked Dec 1 at 9:54
JonMark Perry
9583718
asked Dec 1 at 9:54
JonMark Perry
9583718
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
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 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
add a comment |
1 Answer
1
active
oldest
votes
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 at 10:16
Taras Banakh
15.6k13190
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f316632%2fextension-of-dirichlets-arithmetic-progression-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 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 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 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 at 10:16
Taras Banakh
15.6k13190
edited Dec 1 at 13:26
answered Dec 1 at 10:16
Taras Banakh
15.6k13190
answered Dec 1 at 10:16
Taras Banakh
15.6k13190
answered Dec 1 at 10:16
Taras Banakh
15.6k13190
15.6k13190
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f316632%2fextension-of-dirichlets-arithmetic-progression-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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