Can I find the next prime with sufficient small residue efficiently?
$begingroup$
Suppose, positive integers $N,a,b$ are given.
The object is to determine the smallest prime $p$ larger than $a$ such that $N$ modulo $p$ is smaller than $b$.
Can I determine $p$ efficiently (without just checking all primes until the desired prime is found) ?
Example
$$N=40!$$
$$a=9cdot 10^{15}$$
$$b=10^{10}$$
Brute force gives the result :
? n=40!;a=9*10^15;a=nextprime(a+1);while(n%a>=10^10,a=nextprime(a+1));p=a;print(
p)
9000000017042521
?
Is there a way to speed up the search considerably ?
number-theory elementary-number-theory prime-numbers modular-arithmetic
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
$endgroup$
add a comment |
$begingroup$
Suppose, positive integers $N,a,b$ are given.
The object is to determine the smallest prime $p$ larger than $a$ such that $N$ modulo $p$ is smaller than $b$.
Can I determine $p$ efficiently (without just checking all primes until the desired prime is found) ?
Example
$$N=40!$$
$$a=9cdot 10^{15}$$
$$b=10^{10}$$
Brute force gives the result :
? n=40!;a=9*10^15;a=nextprime(a+1);while(n%a>=10^10,a=nextprime(a+1));p=a;print(
p)
9000000017042521
?
Is there a way to speed up the search considerably ?
number-theory elementary-number-theory prime-numbers modular-arithmetic
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
$endgroup$
$begingroup$
"The object is to determine the smallest prime p larger than a such that N modulo a is smaller than b." I think there is a typo here. Where does $p$ fit in?
$endgroup$
– Mike
Dec 4 '18 at 21:52
$begingroup$
@Mike You are right, fixed.
$endgroup$
– Peter
Dec 4 '18 at 22:14
$begingroup$
I have a hunch chebyshev function can help with this, especially as numbers get larger, but it is nothing more than a hunch.
$endgroup$
– user334732
Dec 4 '18 at 22:35
1
$begingroup$
This is just a question: What about the easier problem of finding an integer (prime or no) $q>a$ s.t. $N$ modulo $q$ is less than $b$
$endgroup$
– Mike
Dec 4 '18 at 23:37
add a comment |
$begingroup$
Suppose, positive integers $N,a,b$ are given.
The object is to determine the smallest prime $p$ larger than $a$ such that $N$ modulo $p$ is smaller than $b$.
Can I determine $p$ efficiently (without just checking all primes until the desired prime is found) ?
Example
$$N=40!$$
$$a=9cdot 10^{15}$$
$$b=10^{10}$$
Brute force gives the result :
? n=40!;a=9*10^15;a=nextprime(a+1);while(n%a>=10^10,a=nextprime(a+1));p=a;print(
p)
9000000017042521
?
Is there a way to speed up the search considerably ?
number-theory elementary-number-theory prime-numbers modular-arithmetic
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
$endgroup$
Suppose, positive integers $N,a,b$ are given.
The object is to determine the smallest prime $p$ larger than $a$ such that $N$ modulo $p$ is smaller than $b$.
Can I determine $p$ efficiently (without just checking all primes until the desired prime is found) ?
Example
$$N=40!$$
$$a=9cdot 10^{15}$$
$$b=10^{10}$$
Brute force gives the result :
? n=40!;a=9*10^15;a=nextprime(a+1);while(n%a>=10^10,a=nextprime(a+1));p=a;print(
p)
9000000017042521
?
Is there a way to speed up the search considerably ?
number-theory elementary-number-theory prime-numbers modular-arithmetic
number-theory elementary-number-theory prime-numbers modular-arithmetic
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
edited Dec 4 '18 at 22:14
Peter
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
asked Dec 4 '18 at 21:34
PeterPeter
46.9k1039125
46.9k1039125
$begingroup$
"The object is to determine the smallest prime p larger than a such that N modulo a is smaller than b." I think there is a typo here. Where does $p$ fit in?
$endgroup$
– Mike
Dec 4 '18 at 21:52
$begingroup$
@Mike You are right, fixed.
$endgroup$
– Peter
Dec 4 '18 at 22:14
$begingroup$
I have a hunch chebyshev function can help with this, especially as numbers get larger, but it is nothing more than a hunch.
$endgroup$
– user334732
Dec 4 '18 at 22:35
1
$begingroup$
This is just a question: What about the easier problem of finding an integer (prime or no) $q>a$ s.t. $N$ modulo $q$ is less than $b$
$endgroup$
– Mike
Dec 4 '18 at 23:37
add a comment |
$begingroup$
"The object is to determine the smallest prime p larger than a such that N modulo a is smaller than b." I think there is a typo here. Where does $p$ fit in?
$endgroup$
– Mike
Dec 4 '18 at 21:52
$begingroup$
@Mike You are right, fixed.
$endgroup$
– Peter
Dec 4 '18 at 22:14
$begingroup$
I have a hunch chebyshev function can help with this, especially as numbers get larger, but it is nothing more than a hunch.
$endgroup$
– user334732
Dec 4 '18 at 22:35
1
$begingroup$
This is just a question: What about the easier problem of finding an integer (prime or no) $q>a$ s.t. $N$ modulo $q$ is less than $b$
$endgroup$
– Mike
Dec 4 '18 at 23:37
$begingroup$
"The object is to determine the smallest prime p larger than a such that N modulo a is smaller than b." I think there is a typo here. Where does $p$ fit in?
$endgroup$
– Mike
Dec 4 '18 at 21:52
$begingroup$
"The object is to determine the smallest prime p larger than a such that N modulo a is smaller than b." I think there is a typo here. Where does $p$ fit in?
$endgroup$
– Mike
Dec 4 '18 at 21:52
$begingroup$
@Mike You are right, fixed.
$endgroup$
– Peter
Dec 4 '18 at 22:14
$begingroup$
@Mike You are right, fixed.
$endgroup$
– Peter
Dec 4 '18 at 22:14
$begingroup$
I have a hunch chebyshev function can help with this, especially as numbers get larger, but it is nothing more than a hunch.
$endgroup$
– user334732
Dec 4 '18 at 22:35
$begingroup$
I have a hunch chebyshev function can help with this, especially as numbers get larger, but it is nothing more than a hunch.
$endgroup$
– user334732
Dec 4 '18 at 22:35
1
1
$begingroup$
This is just a question: What about the easier problem of finding an integer (prime or no) $q>a$ s.t. $N$ modulo $q$ is less than $b$
$endgroup$
– Mike
Dec 4 '18 at 23:37
$begingroup$
This is just a question: What about the easier problem of finding an integer (prime or no) $q>a$ s.t. $N$ modulo $q$ is less than $b$
$endgroup$
– Mike
Dec 4 '18 at 23:37
add a comment |
0
active
oldest
votes
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: "69"
};
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%2fmath.stackexchange.com%2fquestions%2f3026203%2fcan-i-find-the-next-prime-with-sufficient-small-residue-efficiently%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- 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.
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%2fmath.stackexchange.com%2fquestions%2f3026203%2fcan-i-find-the-next-prime-with-sufficient-small-residue-efficiently%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
$begingroup$
"The object is to determine the smallest prime p larger than a such that N modulo a is smaller than b." I think there is a typo here. Where does $p$ fit in?
$endgroup$
– Mike
Dec 4 '18 at 21:52
$begingroup$
@Mike You are right, fixed.
$endgroup$
– Peter
Dec 4 '18 at 22:14
$begingroup$
I have a hunch chebyshev function can help with this, especially as numbers get larger, but it is nothing more than a hunch.
$endgroup$
– user334732
Dec 4 '18 at 22:35
1
$begingroup$
This is just a question: What about the easier problem of finding an integer (prime or no) $q>a$ s.t. $N$ modulo $q$ is less than $b$
$endgroup$
– Mike
Dec 4 '18 at 23:37