How many solutions do $x^{p-1} equiv 1 pmod p$ and $x^{p-1} equiv 2 pmod p$ have?












2












$begingroup$


This is my first post so I apologize for any kind of error.



I'm preparing a magistral degree exam in number theory, and I'm performing some exercise.
I'm asking here this question: how can I prove how many solutions there are for $x^{p-1} equiv 1pmod p$ and $x^{p-1} equiv 2 pmod p$?



Edit: $p$ is an odd prime.






elementary-number-theory modular-arithmetic







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  • $begingroup$
    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    $endgroup$
    – José Carlos Santos
    Nov 26 '18 at 9:15










  • $begingroup$
    @Alessar: You may take example for $p$ say $p=5$ to get an intuitive understanding.
    $endgroup$
    – Yadati Kiran
    Nov 26 '18 at 9:32


















2












$begingroup$


This is my first post so I apologize for any kind of error.



I'm preparing a magistral degree exam in number theory, and I'm performing some exercise.
I'm asking here this question: how can I prove how many solutions there are for $x^{p-1} equiv 1pmod p$ and $x^{p-1} equiv 2 pmod p$?



Edit: $p$ is an odd prime.






elementary-number-theory modular-arithmetic







share|cite|improve this question











$endgroup$












  • $begingroup$
    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    $endgroup$
    – José Carlos Santos
    Nov 26 '18 at 9:15










  • $begingroup$
    @Alessar: You may take example for $p$ say $p=5$ to get an intuitive understanding.
    $endgroup$
    – Yadati Kiran
    Nov 26 '18 at 9:32
















2












2








2





$begingroup$


This is my first post so I apologize for any kind of error.



I'm preparing a magistral degree exam in number theory, and I'm performing some exercise.
I'm asking here this question: how can I prove how many solutions there are for $x^{p-1} equiv 1pmod p$ and $x^{p-1} equiv 2 pmod p$?



Edit: $p$ is an odd prime.






elementary-number-theory modular-arithmetic







share|cite|improve this question











$endgroup$




This is my first post so I apologize for any kind of error.



I'm preparing a magistral degree exam in number theory, and I'm performing some exercise.
I'm asking here this question: how can I prove how many solutions there are for $x^{p-1} equiv 1pmod p$ and $x^{p-1} equiv 2 pmod p$?



Edit: $p$ is an odd prime.






elementary-number-theory modular-arithmetic




elementary-number-theory modular-arithmetic






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 26 '18 at 9:42









Batominovski

1




1










asked Nov 26 '18 at 9:07









AlessarAlessar

27115




27115












  • $begingroup$
    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    $endgroup$
    – José Carlos Santos
    Nov 26 '18 at 9:15










  • $begingroup$
    @Alessar: You may take example for $p$ say $p=5$ to get an intuitive understanding.
    $endgroup$
    – Yadati Kiran
    Nov 26 '18 at 9:32




















  • $begingroup$
    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    $endgroup$
    – José Carlos Santos
    Nov 26 '18 at 9:15










  • $begingroup$
    @Alessar: You may take example for $p$ say $p=5$ to get an intuitive understanding.
    $endgroup$
    – Yadati Kiran
    Nov 26 '18 at 9:32


















$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Nov 26 '18 at 9:15




$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
$endgroup$
– José Carlos Santos
Nov 26 '18 at 9:15












$begingroup$
@Alessar: You may take example for $p$ say $p=5$ to get an intuitive understanding.
$endgroup$
– Yadati Kiran
Nov 26 '18 at 9:32






$begingroup$
@Alessar: You may take example for $p$ say $p=5$ to get an intuitive understanding.
$endgroup$
– Yadati Kiran
Nov 26 '18 at 9:32












1 Answer
1






active

oldest

votes


















2












$begingroup$

Do you know Fermat‘s little theorem?



Consider the multiplicative group $Bbb Z^times_p$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Yes I know it, this is the same as consider x^{p} equiv x(mod p), but for the second questions I'm really stuck
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:31






  • 2




    $begingroup$
    If you know Fermat's Little Theorem, then you would know that, if $p$ is a prime natural number and $x$ is an integer such that $pnmid x$, then $x^{p-1}equiv 1pmod{p}$. Thus, unless $p=2$, $x^{p-1}equiv 2pmod{p}$ has no solution.
    $endgroup$
    – Batominovski
    Nov 26 '18 at 9:40










  • $begingroup$
    Thank you so much, it's the first time I study number theory
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:44











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

Do you know Fermat‘s little theorem?



Consider the multiplicative group $Bbb Z^times_p$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Yes I know it, this is the same as consider x^{p} equiv x(mod p), but for the second questions I'm really stuck
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:31






  • 2




    $begingroup$
    If you know Fermat's Little Theorem, then you would know that, if $p$ is a prime natural number and $x$ is an integer such that $pnmid x$, then $x^{p-1}equiv 1pmod{p}$. Thus, unless $p=2$, $x^{p-1}equiv 2pmod{p}$ has no solution.
    $endgroup$
    – Batominovski
    Nov 26 '18 at 9:40










  • $begingroup$
    Thank you so much, it's the first time I study number theory
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:44
















2












$begingroup$

Do you know Fermat‘s little theorem?



Consider the multiplicative group $Bbb Z^times_p$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Yes I know it, this is the same as consider x^{p} equiv x(mod p), but for the second questions I'm really stuck
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:31






  • 2




    $begingroup$
    If you know Fermat's Little Theorem, then you would know that, if $p$ is a prime natural number and $x$ is an integer such that $pnmid x$, then $x^{p-1}equiv 1pmod{p}$. Thus, unless $p=2$, $x^{p-1}equiv 2pmod{p}$ has no solution.
    $endgroup$
    – Batominovski
    Nov 26 '18 at 9:40










  • $begingroup$
    Thank you so much, it's the first time I study number theory
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:44














2












2








2





$begingroup$

Do you know Fermat‘s little theorem?



Consider the multiplicative group $Bbb Z^times_p$.






share|cite|improve this answer









$endgroup$



Do you know Fermat‘s little theorem?



Consider the multiplicative group $Bbb Z^times_p$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 26 '18 at 9:12









Lukas KoflerLukas Kofler

1,2632519




1,2632519












  • $begingroup$
    Yes I know it, this is the same as consider x^{p} equiv x(mod p), but for the second questions I'm really stuck
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:31






  • 2




    $begingroup$
    If you know Fermat's Little Theorem, then you would know that, if $p$ is a prime natural number and $x$ is an integer such that $pnmid x$, then $x^{p-1}equiv 1pmod{p}$. Thus, unless $p=2$, $x^{p-1}equiv 2pmod{p}$ has no solution.
    $endgroup$
    – Batominovski
    Nov 26 '18 at 9:40










  • $begingroup$
    Thank you so much, it's the first time I study number theory
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:44


















  • $begingroup$
    Yes I know it, this is the same as consider x^{p} equiv x(mod p), but for the second questions I'm really stuck
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:31






  • 2




    $begingroup$
    If you know Fermat's Little Theorem, then you would know that, if $p$ is a prime natural number and $x$ is an integer such that $pnmid x$, then $x^{p-1}equiv 1pmod{p}$. Thus, unless $p=2$, $x^{p-1}equiv 2pmod{p}$ has no solution.
    $endgroup$
    – Batominovski
    Nov 26 '18 at 9:40










  • $begingroup$
    Thank you so much, it's the first time I study number theory
    $endgroup$
    – Alessar
    Nov 26 '18 at 9:44
















$begingroup$
Yes I know it, this is the same as consider x^{p} equiv x(mod p), but for the second questions I'm really stuck
$endgroup$
– Alessar
Nov 26 '18 at 9:31




$begingroup$
Yes I know it, this is the same as consider x^{p} equiv x(mod p), but for the second questions I'm really stuck
$endgroup$
– Alessar
Nov 26 '18 at 9:31




2




2




$begingroup$
If you know Fermat's Little Theorem, then you would know that, if $p$ is a prime natural number and $x$ is an integer such that $pnmid x$, then $x^{p-1}equiv 1pmod{p}$. Thus, unless $p=2$, $x^{p-1}equiv 2pmod{p}$ has no solution.
$endgroup$
– Batominovski
Nov 26 '18 at 9:40




$begingroup$
If you know Fermat's Little Theorem, then you would know that, if $p$ is a prime natural number and $x$ is an integer such that $pnmid x$, then $x^{p-1}equiv 1pmod{p}$. Thus, unless $p=2$, $x^{p-1}equiv 2pmod{p}$ has no solution.
$endgroup$
– Batominovski
Nov 26 '18 at 9:40












$begingroup$
Thank you so much, it's the first time I study number theory
$endgroup$
– Alessar
Nov 26 '18 at 9:44




$begingroup$
Thank you so much, it's the first time I study number theory
$endgroup$
– Alessar
Nov 26 '18 at 9:44


















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