Is there an anomaly in the distribution of Mersenne primes?
$begingroup$
In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that
$$2^{82589933} - 1$$
is the largest known (Mersenne) prime, corresponding to the $51^{text{st}}$ known even perfect number
$$2^{82589932}(2^{82589933} - 1).$$
According to GIMPS' press release:
This is GIMPS' $12^{text{th}}$ prime discovery between $2^{20000000}-1$ and $2^{85000000}-1$, triple the expected number of new primes. One reason to search for new primes is to match actual results with expected results. This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes are incorrect. However, if the trend continues it may be worth further investigation.
Here are my questions:
(1) How is the expected number of new Mersenne primes computed? Is there some underlying statistical model that is used?
(2) What existing theories would be incorrect if there was indeed an anomaly in the distribution of Mersenne primes?
The following question appears to be related but this is not a duplicate of that one.
elementary-number-theory perfect-numbers distribution-of-primes mersenne-numbers computational-number-theory
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
$endgroup$
|
show 5 more comments
$begingroup$
In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that
$$2^{82589933} - 1$$
is the largest known (Mersenne) prime, corresponding to the $51^{text{st}}$ known even perfect number
$$2^{82589932}(2^{82589933} - 1).$$
According to GIMPS' press release:
This is GIMPS' $12^{text{th}}$ prime discovery between $2^{20000000}-1$ and $2^{85000000}-1$, triple the expected number of new primes. One reason to search for new primes is to match actual results with expected results. This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes are incorrect. However, if the trend continues it may be worth further investigation.
Here are my questions:
(1) How is the expected number of new Mersenne primes computed? Is there some underlying statistical model that is used?
(2) What existing theories would be incorrect if there was indeed an anomaly in the distribution of Mersenne primes?
The following question appears to be related but this is not a duplicate of that one.
elementary-number-theory perfect-numbers distribution-of-primes mersenne-numbers computational-number-theory
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
$endgroup$
2
$begingroup$
It is related to the theories derived from conjectures related to Mersenne primes, here is a short article.
$endgroup$
– rtybase
Dec 22 '18 at 9:50
$begingroup$
Thanks for the link, @rtybase! Checking it out now... =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:03
1
$begingroup$
@rtybase, if you could just further expound on your last comment with an actual (not just a link-only) answer, then I will be more than happy to accept it! =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:25
1
$begingroup$
It's Christmas coming, I have to entertain my family :). But if you found the link useful, I am glad I was helpful!
$endgroup$
– rtybase
Dec 22 '18 at 10:38
1
$begingroup$
@JoseArnaldoBebitaDris It is extremely hard to estimate the number of Mersenne primes in a range of huge exponents. Moreover, for events that are very unlikely (for example a huge number to be prime), it is not so unusual that we observe much more events than expected. The formula giving the number of Mersenne numbers will anyway only be a rough guess, we cannot expect that it is "accurate" , in particular the formula cannot prove the conjecture that infinite many Mersenne primes exist.
$endgroup$
– Peter
Dec 22 '18 at 13:10
|
show 5 more comments
$begingroup$
In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that
$$2^{82589933} - 1$$
is the largest known (Mersenne) prime, corresponding to the $51^{text{st}}$ known even perfect number
$$2^{82589932}(2^{82589933} - 1).$$
According to GIMPS' press release:
This is GIMPS' $12^{text{th}}$ prime discovery between $2^{20000000}-1$ and $2^{85000000}-1$, triple the expected number of new primes. One reason to search for new primes is to match actual results with expected results. This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes are incorrect. However, if the trend continues it may be worth further investigation.
Here are my questions:
(1) How is the expected number of new Mersenne primes computed? Is there some underlying statistical model that is used?
(2) What existing theories would be incorrect if there was indeed an anomaly in the distribution of Mersenne primes?
The following question appears to be related but this is not a duplicate of that one.
elementary-number-theory perfect-numbers distribution-of-primes mersenne-numbers computational-number-theory
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
$endgroup$
In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that
$$2^{82589933} - 1$$
is the largest known (Mersenne) prime, corresponding to the $51^{text{st}}$ known even perfect number
$$2^{82589932}(2^{82589933} - 1).$$
According to GIMPS' press release:
This is GIMPS' $12^{text{th}}$ prime discovery between $2^{20000000}-1$ and $2^{85000000}-1$, triple the expected number of new primes. One reason to search for new primes is to match actual results with expected results. This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes are incorrect. However, if the trend continues it may be worth further investigation.
Here are my questions:
(1) How is the expected number of new Mersenne primes computed? Is there some underlying statistical model that is used?
(2) What existing theories would be incorrect if there was indeed an anomaly in the distribution of Mersenne primes?
The following question appears to be related but this is not a duplicate of that one.
elementary-number-theory perfect-numbers distribution-of-primes mersenne-numbers computational-number-theory
elementary-number-theory perfect-numbers distribution-of-primes mersenne-numbers computational-number-theory
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
edited Dec 23 '18 at 6:33
Jose Arnaldo Bebita Dris
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
asked Dec 22 '18 at 8:57
Jose Arnaldo Bebita DrisJose Arnaldo Bebita Dris
5,43641944
5,43641944
2
$begingroup$
It is related to the theories derived from conjectures related to Mersenne primes, here is a short article.
$endgroup$
– rtybase
Dec 22 '18 at 9:50
$begingroup$
Thanks for the link, @rtybase! Checking it out now... =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:03
1
$begingroup$
@rtybase, if you could just further expound on your last comment with an actual (not just a link-only) answer, then I will be more than happy to accept it! =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:25
1
$begingroup$
It's Christmas coming, I have to entertain my family :). But if you found the link useful, I am glad I was helpful!
$endgroup$
– rtybase
Dec 22 '18 at 10:38
1
$begingroup$
@JoseArnaldoBebitaDris It is extremely hard to estimate the number of Mersenne primes in a range of huge exponents. Moreover, for events that are very unlikely (for example a huge number to be prime), it is not so unusual that we observe much more events than expected. The formula giving the number of Mersenne numbers will anyway only be a rough guess, we cannot expect that it is "accurate" , in particular the formula cannot prove the conjecture that infinite many Mersenne primes exist.
$endgroup$
– Peter
Dec 22 '18 at 13:10
|
show 5 more comments
2
$begingroup$
It is related to the theories derived from conjectures related to Mersenne primes, here is a short article.
$endgroup$
– rtybase
Dec 22 '18 at 9:50
$begingroup$
Thanks for the link, @rtybase! Checking it out now... =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:03
1
$begingroup$
@rtybase, if you could just further expound on your last comment with an actual (not just a link-only) answer, then I will be more than happy to accept it! =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:25
1
$begingroup$
It's Christmas coming, I have to entertain my family :). But if you found the link useful, I am glad I was helpful!
$endgroup$
– rtybase
Dec 22 '18 at 10:38
1
$begingroup$
@JoseArnaldoBebitaDris It is extremely hard to estimate the number of Mersenne primes in a range of huge exponents. Moreover, for events that are very unlikely (for example a huge number to be prime), it is not so unusual that we observe much more events than expected. The formula giving the number of Mersenne numbers will anyway only be a rough guess, we cannot expect that it is "accurate" , in particular the formula cannot prove the conjecture that infinite many Mersenne primes exist.
$endgroup$
– Peter
Dec 22 '18 at 13:10
2
2
$begingroup$
It is related to the theories derived from conjectures related to Mersenne primes, here is a short article.
$endgroup$
– rtybase
Dec 22 '18 at 9:50
$begingroup$
It is related to the theories derived from conjectures related to Mersenne primes, here is a short article.
$endgroup$
– rtybase
Dec 22 '18 at 9:50
$begingroup$
Thanks for the link, @rtybase! Checking it out now... =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:03
$begingroup$
Thanks for the link, @rtybase! Checking it out now... =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:03
1
1
$begingroup$
@rtybase, if you could just further expound on your last comment with an actual (not just a link-only) answer, then I will be more than happy to accept it! =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:25
$begingroup$
@rtybase, if you could just further expound on your last comment with an actual (not just a link-only) answer, then I will be more than happy to accept it! =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:25
1
1
$begingroup$
It's Christmas coming, I have to entertain my family :). But if you found the link useful, I am glad I was helpful!
$endgroup$
– rtybase
Dec 22 '18 at 10:38
$begingroup$
It's Christmas coming, I have to entertain my family :). But if you found the link useful, I am glad I was helpful!
$endgroup$
– rtybase
Dec 22 '18 at 10:38
1
1
$begingroup$
@JoseArnaldoBebitaDris It is extremely hard to estimate the number of Mersenne primes in a range of huge exponents. Moreover, for events that are very unlikely (for example a huge number to be prime), it is not so unusual that we observe much more events than expected. The formula giving the number of Mersenne numbers will anyway only be a rough guess, we cannot expect that it is "accurate" , in particular the formula cannot prove the conjecture that infinite many Mersenne primes exist.
$endgroup$
– Peter
Dec 22 '18 at 13:10
$begingroup$
@JoseArnaldoBebitaDris It is extremely hard to estimate the number of Mersenne primes in a range of huge exponents. Moreover, for events that are very unlikely (for example a huge number to be prime), it is not so unusual that we observe much more events than expected. The formula giving the number of Mersenne numbers will anyway only be a rough guess, we cannot expect that it is "accurate" , in particular the formula cannot prove the conjecture that infinite many Mersenne primes exist.
$endgroup$
– Peter
Dec 22 '18 at 13:10
|
show 5 more comments
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%2f3049234%2fis-there-an-anomaly-in-the-distribution-of-mersenne-primes%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%2f3049234%2fis-there-an-anomaly-in-the-distribution-of-mersenne-primes%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
$begingroup$
It is related to the theories derived from conjectures related to Mersenne primes, here is a short article.
$endgroup$
– rtybase
Dec 22 '18 at 9:50
$begingroup$
Thanks for the link, @rtybase! Checking it out now... =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:03
1
$begingroup$
@rtybase, if you could just further expound on your last comment with an actual (not just a link-only) answer, then I will be more than happy to accept it! =)
$endgroup$
– Jose Arnaldo Bebita Dris
Dec 22 '18 at 10:25
1
$begingroup$
It's Christmas coming, I have to entertain my family :). But if you found the link useful, I am glad I was helpful!
$endgroup$
– rtybase
Dec 22 '18 at 10:38
1
$begingroup$
@JoseArnaldoBebitaDris It is extremely hard to estimate the number of Mersenne primes in a range of huge exponents. Moreover, for events that are very unlikely (for example a huge number to be prime), it is not so unusual that we observe much more events than expected. The formula giving the number of Mersenne numbers will anyway only be a rough guess, we cannot expect that it is "accurate" , in particular the formula cannot prove the conjecture that infinite many Mersenne primes exist.
$endgroup$
– Peter
Dec 22 '18 at 13:10