Upper bound on the number of integer solutions to $y^p=x^2+2$, where $p$ is prime The Next CEO of Stack Overflowthe number of integer solutions to $y^p = x^2 +4$What Legendre's Conjecture Implies About the Upper Bound For the Prime Gap Above Any Natural NumberTheorem: Odd positive integer N is a prime number if …the number of integer solutions to $y^p = x^2 +4$Upper bound of the number of solutions of a diophantine equation.Proving the equation $m^2+m+1=n(16n^2-12n+3)$ has only two integer solutions.Find positive integer solutions of the Diophantine equation $x^4+py^4=z^2$.Upper bound to the number of solutions of the following Diophantine Equation.Find all integer solutions to the equation $x^2 − x = y^5 − y$.Is there an upper bound to the number of weight $2$ newforms for any level $N$?Showing that $x^2+5=y^3$ has no integer solutions.
Customer Requests (Sometimes) Drive Me Bonkers!
Why didn't Khan get resurrected in the Genesis Explosion?
How to use tikz in fbox?
Only print output after finding pattern
How to be diplomatic in refusing to write code that breaches the privacy of our users
How do scammers retract money, while you can’t?
How to write papers efficiently when English isn't my first language?
How did people program for Consoles with multiple CPUs?
How to Reset Passwords on Multiple Websites Easily?
I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin
Implement the Thanos sorting algorithm
Science fiction (dystopian) short story set after WWIII
How do I go from 300 unfinished/half written blog posts, to published posts?
Should I tutor a student who I know has cheated on their homework?
When airplanes disconnect from a tanker during air to air refueling, why do they bank so sharply to the right?
Natural language into sentence logic
Why did we only see the N-1 starfighters in one film?
Why doesn't a table tennis ball float on the surface? How do we calculate buoyancy here?
What is the purpose of the Evocation wizard's Potent Cantrip feature?
How do I construct this japanese bowl?
How can I quit an app using Terminal?
How to get regions to plot as graphics
Shade part of a Venn diagram
Grabbing quick drinks
Upper bound on the number of integer solutions to $y^p=x^2+2$, where $p$ is prime
The Next CEO of Stack Overflowthe number of integer solutions to $y^p = x^2 +4$What Legendre's Conjecture Implies About the Upper Bound For the Prime Gap Above Any Natural NumberTheorem: Odd positive integer N is a prime number if …the number of integer solutions to $y^p = x^2 +4$Upper bound of the number of solutions of a diophantine equation.Proving the equation $m^2+m+1=n(16n^2-12n+3)$ has only two integer solutions.Find positive integer solutions of the Diophantine equation $x^4+py^4=z^2$.Upper bound to the number of solutions of the following Diophantine Equation.Find all integer solutions to the equation $x^2 − x = y^5 − y$.Is there an upper bound to the number of weight $2$ newforms for any level $N$?Showing that $x^2+5=y^3$ has no integer solutions.
$begingroup$
I need to find an upper bound on the number of solutions of the Diophantine equation $y^p=x^2+2$, where $p$ is prime.
I have previously considered the equation $y^3=x^2+2$ and proved its solutions are $(5,3)$ and $(-5,3)$ by factorizing it as $y^3=(x-sqrt-2)(x+sqrt-2)$ and showing that $(x-sqrt-2)$ and $(x+sqrt-2)$ are relatively prime by properties of the UFD $mathbbZ[sqrt-2]$ and then showing that $x+sqrt-2$ is a cube.
In the last stept, I have used the binomial theorem to find the solutions:
$x+sqrt-2=(a+bsqrt-2)^3$, where $a,b in mathbbZ$.
I think I need to do something similar for $y^p=x^2+2$. Obviously, I can show that $x+sqrt-2$ is a $p$th power and then use the binomial theorem to get:
$x+sqrt-2=(a+bsqrt-2)^p=sum_k=0^p binompk a^p-k (bsqrt-2)^k$.
How do I obtain an upper bound on the number of solutions of the equation from here?
number-theory diophantine-equations
asked Mar 18 at 12:53
vladr10vladr10
423
$endgroup$
add a comment |
$begingroup$
I need to find an upper bound on the number of solutions of the Diophantine equation $y^p=x^2+2$, where $p$ is prime.
I have previously considered the equation $y^3=x^2+2$ and proved its solutions are $(5,3)$ and $(-5,3)$ by factorizing it as $y^3=(x-sqrt-2)(x+sqrt-2)$ and showing that $(x-sqrt-2)$ and $(x+sqrt-2)$ are relatively prime by properties of the UFD $mathbbZ[sqrt-2]$ and then showing that $x+sqrt-2$ is a cube.
In the last stept, I have used the binomial theorem to find the solutions:
$x+sqrt-2=(a+bsqrt-2)^3$, where $a,b in mathbbZ$.
I think I need to do something similar for $y^p=x^2+2$. Obviously, I can show that $x+sqrt-2$ is a $p$th power and then use the binomial theorem to get:
$x+sqrt-2=(a+bsqrt-2)^p=sum_k=0^p binompk a^p-k (bsqrt-2)^k$.
How do I obtain an upper bound on the number of solutions of the equation from here?
number-theory diophantine-equations
asked Mar 18 at 12:53
vladr10vladr10
423
$endgroup$
add a comment |
$begingroup$
I need to find an upper bound on the number of solutions of the Diophantine equation $y^p=x^2+2$, where $p$ is prime.
I have previously considered the equation $y^3=x^2+2$ and proved its solutions are $(5,3)$ and $(-5,3)$ by factorizing it as $y^3=(x-sqrt-2)(x+sqrt-2)$ and showing that $(x-sqrt-2)$ and $(x+sqrt-2)$ are relatively prime by properties of the UFD $mathbbZ[sqrt-2]$ and then showing that $x+sqrt-2$ is a cube.
In the last stept, I have used the binomial theorem to find the solutions:
$x+sqrt-2=(a+bsqrt-2)^3$, where $a,b in mathbbZ$.
I think I need to do something similar for $y^p=x^2+2$. Obviously, I can show that $x+sqrt-2$ is a $p$th power and then use the binomial theorem to get:
$x+sqrt-2=(a+bsqrt-2)^p=sum_k=0^p binompk a^p-k (bsqrt-2)^k$.
How do I obtain an upper bound on the number of solutions of the equation from here?
number-theory diophantine-equations
asked Mar 18 at 12:53
vladr10vladr10
423
$endgroup$
I need to find an upper bound on the number of solutions of the Diophantine equation $y^p=x^2+2$, where $p$ is prime.
I have previously considered the equation $y^3=x^2+2$ and proved its solutions are $(5,3)$ and $(-5,3)$ by factorizing it as $y^3=(x-sqrt-2)(x+sqrt-2)$ and showing that $(x-sqrt-2)$ and $(x+sqrt-2)$ are relatively prime by properties of the UFD $mathbbZ[sqrt-2]$ and then showing that $x+sqrt-2$ is a cube.
In the last stept, I have used the binomial theorem to find the solutions:
$x+sqrt-2=(a+bsqrt-2)^3$, where $a,b in mathbbZ$.
I think I need to do something similar for $y^p=x^2+2$. Obviously, I can show that $x+sqrt-2$ is a $p$th power and then use the binomial theorem to get:
$x+sqrt-2=(a+bsqrt-2)^p=sum_k=0^p binompk a^p-k (bsqrt-2)^k$.
How do I obtain an upper bound on the number of solutions of the equation from here?
number-theory diophantine-equations
number-theory diophantine-equations
asked Mar 18 at 12:53
vladr10vladr10
423
asked Mar 18 at 12:53
vladr10vladr10
423
asked Mar 18 at 12:53
vladr10vladr10
423
asked Mar 18 at 12:53
vladr10vladr10
423
asked Mar 18 at 12:53
vladr10vladr10
423
423
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This has been discussed in the article here on the Diophantine equation
$$
y^n=x^2+c
$$
for a integers $n$ and $c$, so in particualr for $c=2$ and $n=p$ prime.
Ljunggren [20] generalised Fermat’s result and proved that for
$c = 2$ the equation has no solution other than $x = 5$.
Other references on this site: the number of integer solutions to $y^p = x^2 +4$
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
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: "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%2f3152746%2fupper-bound-on-the-number-of-integer-solutions-to-yp-x22-where-p-is-prim%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
$begingroup$
This has been discussed in the article here on the Diophantine equation
$$
y^n=x^2+c
$$
for a integers $n$ and $c$, so in particualr for $c=2$ and $n=p$ prime.
Ljunggren [20] generalised Fermat’s result and proved that for
$c = 2$ the equation has no solution other than $x = 5$.
Other references on this site: the number of integer solutions to $y^p = x^2 +4$
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
add a comment |
$begingroup$
This has been discussed in the article here on the Diophantine equation
$$
y^n=x^2+c
$$
for a integers $n$ and $c$, so in particualr for $c=2$ and $n=p$ prime.
Ljunggren [20] generalised Fermat’s result and proved that for
$c = 2$ the equation has no solution other than $x = 5$.
Other references on this site: the number of integer solutions to $y^p = x^2 +4$
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
add a comment |
$begingroup$
This has been discussed in the article here on the Diophantine equation
$$
y^n=x^2+c
$$
for a integers $n$ and $c$, so in particualr for $c=2$ and $n=p$ prime.
Ljunggren [20] generalised Fermat’s result and proved that for
$c = 2$ the equation has no solution other than $x = 5$.
Other references on this site: the number of integer solutions to $y^p = x^2 +4$
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
$endgroup$
This has been discussed in the article here on the Diophantine equation
$$
y^n=x^2+c
$$
for a integers $n$ and $c$, so in particualr for $c=2$ and $n=p$ prime.
Ljunggren [20] generalised Fermat’s result and proved that for
$c = 2$ the equation has no solution other than $x = 5$.
Other references on this site: the number of integer solutions to $y^p = x^2 +4$
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
edited Mar 18 at 13:19
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
answered Mar 18 at 12:59
Dietrich BurdeDietrich Burde
81.5k648106
81.5k648106
add a comment |
add a comment |
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%2f3152746%2fupper-bound-on-the-number-of-integer-solutions-to-yp-x22-where-p-is-prim%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
