Fdtxjr

Life insurance that covers only simultaneous/dual deaths

Why are there 40 737 Max planes in flight when they have been grounded as not airworthy?

At what level can a dragon innately cast its spells?

What options are left, if Britain cannot decide?

Official degrees of earth’s rotation per day

How to deal with taxi scam when on vacation?

Is having access to past exams cheating and, if yes, could it be proven just by a good grade?

Be in awe of my brilliance!

How to explain that I do not want to visit a country due to personal safety concern?

Why do Australian milk farmers need to protest supermarkets' milk price?

Bash: What does "masking return values" mean?

What is a good source for large tables on the properties of water?

PTIJ: Who should pay for Uber rides: the child or the parent?

Why using two cd commands in bash script does not execute the second command

Does the statement `int val = (++i > ++j) ? ++i : ++j;` invoke undefined behavior?

When do we add an hyphen (-) to a complex adjective word?

Employee lack of ownership

How is the Swiss post e-voting system supposed to work, and how was it wrong?

How do I hide Chekhov's Gun?

Happy pi day, everyone!

It's a yearly task, alright

An Accountant Seeks the Help of a Mathematician

Does this AnyDice function accurately calculate the number of ogres you make unconcious with three 4th-level castings of Sleep?

Instead of Universal Basic Income, why not Universal Basic NEEDS?

Question about Wilson's theorem

Help manipulating Wilson's Theorem.Wilson's Theorem and related identitiesWilson's theoremPart of a proof for Wilson's TheoremWilson's theorem intuition(Elementary Number Theory) Using Wilson's Theorem ..??Using Wilson's Theorem to prove Fermat's Little TheoremProve $717$ is not prime using Wilson's TheoremProving Wilson's theoremQuestion about Wilson's theorem, when $n = 4$.

1

$begingroup$

For Wilson's theorem, if $p$ is prime then $(p-1) equiv -1 modp$ and if not $0 modp$ except for $p=4$, is $p$ and integer or natural number?

Studying Wilson's theorem for Double, Hyper, Sub and Double factorials and I have begun by stating what Wilson's theorem actually is.

However is $p in mathbbZ$? Or is $p in mathbbN$?

abstract-algebra elementary-number-theory

share|cite|improve this question

edited Mar 11 at 9:55

Andrews

1,2691421

asked Mar 11 at 9:45

J.BryJ.Bry

486

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  It's $(p-1)!$ : how do you define factorials for negative integers ?
  $endgroup$
  – Max
  Mar 11 at 10:08
  

  
  
  
  

add a comment | 

1

$begingroup$

For Wilson's theorem, if $p$ is prime then $(p-1) equiv -1 modp$ and if not $0 modp$ except for $p=4$, is $p$ and integer or natural number?

Studying Wilson's theorem for Double, Hyper, Sub and Double factorials and I have begun by stating what Wilson's theorem actually is.

However is $p in mathbbZ$? Or is $p in mathbbN$?

abstract-algebra elementary-number-theory

share|cite|improve this question

edited Mar 11 at 9:55

Andrews

1,2691421

asked Mar 11 at 9:45

J.BryJ.Bry

486

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  It's $(p-1)!$ : how do you define factorials for negative integers ?
  $endgroup$
  – Max
  Mar 11 at 10:08
  

  
  
  
  

add a comment | 

$begingroup$

For Wilson's theorem, if $p$ is prime then $(p-1) equiv -1 modp$ and if not $0 modp$ except for $p=4$, is $p$ and integer or natural number?

Studying Wilson's theorem for Double, Hyper, Sub and Double factorials and I have begun by stating what Wilson's theorem actually is.

However is $p in mathbbZ$? Or is $p in mathbbN$?

abstract-algebra elementary-number-theory

share|cite|improve this question

edited Mar 11 at 9:55

Andrews

1,2691421

asked Mar 11 at 9:45

J.BryJ.Bry

486

$endgroup$

share|cite|improve this question

edited Mar 11 at 9:55

Andrews

1,2691421

asked Mar 11 at 9:45

J.BryJ.Bry

486

share|cite|improve this question

edited Mar 11 at 9:55

Andrews

1,2691421

asked Mar 11 at 9:45

J.BryJ.Bry

486

edited Mar 11 at 9:55

Andrews

1,2691421

edited Mar 11 at 9:55

Andrews

1,2691421

asked Mar 11 at 9:45

J.BryJ.Bry

486

asked Mar 11 at 9:45

J.BryJ.Bry

486

1 Answer
1

active

oldest

votes

2

$begingroup$

I assume you mean $(p-1)!$

It is not a loss of generality to assume $p in mathbbN$, as primes are unique upto multiplication by units (in this case $pm 1$). The factorial function is not defined at negative integers. However, you may restate the theorem for $-p$ as:

$(-1)(-2)...(-p+1)$ is congruent to $-1$ modulo $-p$ if $-p$ is prime and $0$ otherwise (except $-p = -4$).

More specifically, $p$ and $-p$ generate the same ideal in $mathbbZ$, so the quotient fields are isomorphic.

share|cite|improve this answer

answered Mar 11 at 10:06

nammienammie

1899

New contributor

nammie is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3143486%2fquestion-about-wilsons-theorem%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

2

$begingroup$

I assume you mean $(p-1)!$

It is not a loss of generality to assume $p in mathbbN$, as primes are unique upto multiplication by units (in this case $pm 1$). The factorial function is not defined at negative integers. However, you may restate the theorem for $-p$ as:

$(-1)(-2)...(-p+1)$ is congruent to $-1$ modulo $-p$ if $-p$ is prime and $0$ otherwise (except $-p = -4$).

More specifically, $p$ and $-p$ generate the same ideal in $mathbbZ$, so the quotient fields are isomorphic.

share|cite|improve this answer

answered Mar 11 at 10:06

nammienammie

1899

New contributor

nammie is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

2

$begingroup$

I assume you mean $(p-1)!$

It is not a loss of generality to assume $p in mathbbN$, as primes are unique upto multiplication by units (in this case $pm 1$). The factorial function is not defined at negative integers. However, you may restate the theorem for $-p$ as:

$(-1)(-2)...(-p+1)$ is congruent to $-1$ modulo $-p$ if $-p$ is prime and $0$ otherwise (except $-p = -4$).

More specifically, $p$ and $-p$ generate the same ideal in $mathbbZ$, so the quotient fields are isomorphic.

share|cite|improve this answer

answered Mar 11 at 10:06

nammienammie

1899

New contributor

nammie is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

I assume you mean $(p-1)!$

It is not a loss of generality to assume $p in mathbbN$, as primes are unique upto multiplication by units (in this case $pm 1$). The factorial function is not defined at negative integers. However, you may restate the theorem for $-p$ as:

$(-1)(-2)...(-p+1)$ is congruent to $-1$ modulo $-p$ if $-p$ is prime and $0$ otherwise (except $-p = -4$).

More specifically, $p$ and $-p$ generate the same ideal in $mathbbZ$, so the quotient fields are isomorphic.

share|cite|improve this answer

answered Mar 11 at 10:06

nammienammie

1899

New contributor

nammie is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this answer

answered Mar 11 at 10:06

nammienammie

1899

New contributor

nammie is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

answered Mar 11 at 10:06

nammienammie

1899

New contributor

nammie is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

answered Mar 11 at 10:06

nammienammie

1899