Prime number and Divisors












2














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










share|cite|improve this question




















  • 4




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
















2














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










share|cite|improve this question




















  • 4




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














2












2








2


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










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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 48 mins ago









Chinnapparaj R

5,2271826




5,2271826










asked 1 hour ago









Mohammad Mizanur Rahaman

135




135








  • 4




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














  • 4




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








4




4




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




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










1 Answer
1






active

oldest

votes


















5














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
    54 mins ago






  • 1




    @MohammadMizanurRahaman: yes, $p=q=2$
    – Ross Millikan
    44 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









5














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
    54 mins ago






  • 1




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
















5














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
    54 mins ago






  • 1




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














5












5








5






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 1 hour ago









Barycentric_Bash

38728




38728












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






  • 1




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


















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






  • 1




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
















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




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




1




1




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




@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
44 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

Ellipse (mathématiques)

Quarter-circle Tiles

Mont Emei