Near-Legendre Conjecture
$begingroup$
Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$
Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open.
What, if anything, is known about the existence of primes in the intervals
$$
[n^{2+epsilon},(n+1)^{2+epsilon}],
$$
for $n$ large enough?
nt.number-theory prime-numbers
asked 3 hours ago
kodlukodlu
3,72121728
$endgroup$
add a comment |
$begingroup$
Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$
Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open.
What, if anything, is known about the existence of primes in the intervals
$$
[n^{2+epsilon},(n+1)^{2+epsilon}],
$$
for $n$ large enough?
nt.number-theory prime-numbers
asked 3 hours ago
kodlukodlu
3,72121728
$endgroup$
add a comment |
$begingroup$
Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$
Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open.
What, if anything, is known about the existence of primes in the intervals
$$
[n^{2+epsilon},(n+1)^{2+epsilon}],
$$
for $n$ large enough?
nt.number-theory prime-numbers
asked 3 hours ago
kodlukodlu
3,72121728
$endgroup$
Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$
Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open.
What, if anything, is known about the existence of primes in the intervals
$$
[n^{2+epsilon},(n+1)^{2+epsilon}],
$$
for $n$ large enough?
nt.number-theory prime-numbers
nt.number-theory prime-numbers
asked 3 hours ago
kodlukodlu
3,72121728
asked 3 hours ago
kodlukodlu
3,72121728
asked 3 hours ago
kodlukodlu
3,72121728
asked 3 hours ago
kodlukodlu
3,72121728
asked 3 hours ago
kodlukodlu
3,72121728
3,72121728
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Baker, Harman and Pintz showed in 2001 that for $theta=0.525$, the interval $(x,x+x^theta)$ contains at least one prime provided $x$ is sufficiently large. This is equivalent to the interval $[n^{2+epsilon},(n+1)^{2+epsilon}]$ to contain a prime for $epsilon=frac{2theta-1}{1-theta}approx 0.1053$ and $n$ sufficiently large
answered 1 hour ago
SevaSeva
12.6k137102
$endgroup$
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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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%2f321085%2fnear-legendre-conjecture%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
$begingroup$
Baker, Harman and Pintz showed in 2001 that for $theta=0.525$, the interval $(x,x+x^theta)$ contains at least one prime provided $x$ is sufficiently large. This is equivalent to the interval $[n^{2+epsilon},(n+1)^{2+epsilon}]$ to contain a prime for $epsilon=frac{2theta-1}{1-theta}approx 0.1053$ and $n$ sufficiently large
answered 1 hour ago
SevaSeva
12.6k137102
$endgroup$
add a comment |
$begingroup$
Baker, Harman and Pintz showed in 2001 that for $theta=0.525$, the interval $(x,x+x^theta)$ contains at least one prime provided $x$ is sufficiently large. This is equivalent to the interval $[n^{2+epsilon},(n+1)^{2+epsilon}]$ to contain a prime for $epsilon=frac{2theta-1}{1-theta}approx 0.1053$ and $n$ sufficiently large
answered 1 hour ago
SevaSeva
12.6k137102
$endgroup$
add a comment |
$begingroup$
Baker, Harman and Pintz showed in 2001 that for $theta=0.525$, the interval $(x,x+x^theta)$ contains at least one prime provided $x$ is sufficiently large. This is equivalent to the interval $[n^{2+epsilon},(n+1)^{2+epsilon}]$ to contain a prime for $epsilon=frac{2theta-1}{1-theta}approx 0.1053$ and $n$ sufficiently large
answered 1 hour ago
SevaSeva
12.6k137102
$endgroup$
Baker, Harman and Pintz showed in 2001 that for $theta=0.525$, the interval $(x,x+x^theta)$ contains at least one prime provided $x$ is sufficiently large. This is equivalent to the interval $[n^{2+epsilon},(n+1)^{2+epsilon}]$ to contain a prime for $epsilon=frac{2theta-1}{1-theta}approx 0.1053$ and $n$ sufficiently large
answered 1 hour ago
SevaSeva
12.6k137102
answered 1 hour ago
SevaSeva
12.6k137102
answered 1 hour ago
SevaSeva
12.6k137102
answered 1 hour ago
SevaSeva
12.6k137102
12.6k137102
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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%2f321085%2fnear-legendre-conjecture%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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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
