Prime number and Divisors












1














Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?



I came across this question in a Math Olympiad Competition and had no idea how to solve it






elementary-number-theory prime-numbers







share|cite|improve this question




















  • 4




    A number with an odd number of divisors must be square.
    – Lord Shark the Unknown
    29 mins ago
















1














Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?



I came across this question in a Math Olympiad Competition and had no idea how to solve it






elementary-number-theory prime-numbers







share|cite|improve this question




















  • 4




    A number with an odd number of divisors must be square.
    – Lord Shark the Unknown
    29 mins ago














1












1








1


1





Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?



I came across this question in a Math Olympiad Competition and had no idea how to solve it






elementary-number-theory prime-numbers







share|cite|improve this question















Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?



I came across this question in a Math Olympiad Competition and had no idea how to solve it






elementary-number-theory prime-numbers




elementary-number-theory prime-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 12 mins ago









Chinnapparaj R

5,2271826




5,2271826










asked 31 mins ago









Mohammad Mizanur Rahaman

155




155








  • 4




    A number with an odd number of divisors must be square.
    – Lord Shark the Unknown
    29 mins ago














  • 4




    A number with an odd number of divisors must be square.
    – Lord Shark the Unknown
    29 mins ago








4




4




A number with an odd number of divisors must be square.
– Lord Shark the Unknown
29 mins ago




A number with an odd number of divisors must be square.
– Lord Shark the Unknown
29 mins ago










1 Answer
1






active

oldest

votes


















4














Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:



$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$



Then do some casework on it.






share|cite|improve this answer





















  • The answer I have got is 2(maximum value of P). I think I am right ?
    – Mohammad Mizanur Rahaman
    18 mins ago






  • 1




    @MohammadMizanurRahaman: yes, $p=q=2$
    – Ross Millikan
    7 mins ago











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051874%2fprime-number-and-divisors%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









4














Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:



$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$



Then do some casework on it.






share|cite|improve this answer





















  • The answer I have got is 2(maximum value of P). I think I am right ?
    – Mohammad Mizanur Rahaman
    18 mins ago






  • 1




    @MohammadMizanurRahaman: yes, $p=q=2$
    – Ross Millikan
    7 mins ago
















4














Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:



$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$



Then do some casework on it.






share|cite|improve this answer





















  • The answer I have got is 2(maximum value of P). I think I am right ?
    – Mohammad Mizanur Rahaman
    18 mins ago






  • 1




    @MohammadMizanurRahaman: yes, $p=q=2$
    – Ross Millikan
    7 mins ago














4












4








4






Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:



$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$



Then do some casework on it.






share|cite|improve this answer












Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:



$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$



Then do some casework on it.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 26 mins ago









Barycentric_Bash

37728




37728












  • The answer I have got is 2(maximum value of P). I think I am right ?
    – Mohammad Mizanur Rahaman
    18 mins ago






  • 1




    @MohammadMizanurRahaman: yes, $p=q=2$
    – Ross Millikan
    7 mins ago


















  • The answer I have got is 2(maximum value of P). I think I am right ?
    – Mohammad Mizanur Rahaman
    18 mins ago






  • 1




    @MohammadMizanurRahaman: yes, $p=q=2$
    – Ross Millikan
    7 mins ago
















The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
18 mins ago




The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
18 mins ago




1




1




@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
7 mins ago




@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
7 mins ago


















draft saved

draft discarded




















































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.





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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051874%2fprime-number-and-divisors%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

A CLEAN and SIMPLE way to add appendices to Table of Contents and bookmarks

Image
0 I found a lot of questions abount appendices and ToC. Many users want appendices to be grouped in an Appendix part, however some problems arise with ToC, hyperref, PDF viewer bookmarks, and so on. There are different solutions which require extra packages, command patching and other extra code, however none of them satisfies me. I almost found an easy way to accomplish a good result, where appendices are added to bookmarks in the right way and hyperref links point to the right page. However, the number of the "Appendix" part page is wrong (it's the number of appendix A). Is there any EASY way to fix that? This is a MWE: documentclass{book} usepackage[nottoc,notlot,notlof]{tocbibind} usepackage{hyperref} begin{document} frontmatter tableofcontents mainmatter part{First} chapter{...
Read more

Calculate evaluation metrics using cross_val_predict sklearn

Image
1 In the sklearn.model_selection.cross_val_predict page it is stated: Generate cross-validated estimates for each input data point. It is not appropriate to pass these predictions into an evaluation metric. Can someone explain what does it mean? If this gives estimate of Y (y prediction) for every Y (true Y), why can't I calculate metrics such as RMSE or coefficient of determination using these results? python scikit-learn cross-validation share | improve this question edited Nov 28 '18 at 17:52 desertnaut 20.3k 7 43 79 ...
Read more

Insert data from modal to MySQL (multiple modal on website)

Image
0 I have problem with inserting data to MySQL from modal. My modal: <a href="#" class="badge badge-pill badge-success">6 komentarzy</a> <a href="#" class="badge badge-pill badge-danger">brak komentarzy</a> <a data-toggle="modal" href="#add_desk_comm_{$desk_['desk_id']}" data-target="#add_desk_comm_{$desk_['desk_id']}" class="ediiit">(dodaj)</a> <div class="modal fade" id="add_desk_comm_{$desk_['desk_id']}" tabindex="-1" role="dialog" aria-labelledby="edit_printer" aria-hidden="true"> <div class="modal-dialog" role="document"> ...
Read more