Reference Request – $ n $ is a positive prime if $ k ^ {n-1} 1 $ mod $ n $ equals $ 1

When I read the little sentence from Fermat, I found a problem. After that, I realized that the problem is the Chinese Hypothesis.

When I read Wilson's sentence, I found that $ (p-2)! equiv -1 $ mod $ p $ but after that I knew that this is a special case of the Sylow sentences.

I can not prove, can not verify by my computer the following statement, because the Chinese hypothesis is up to true $ n = {340} $ With $ k $ is constant $ 2 $, Here I leave $ 1 <k <n $

Is it true that if $ n $ is a positive prime, if only if $ k ^ {n-1} equiv 1 $ mod $ n $ With $ 1 <k <n $

  • If $ n $ is a positive prime after the small Fermat theorem $ Rightarrow $ $ k ^ {n-1} equiv 1 $ mod $ n $,

  • If $ k ^ {n-1} equiv 1 $ mod $ n $ With $ k = 1, 2, 3, …, n-1 $ then $ n $ is also prime.

See also:

nt.number theory – For a problem where $ frac { text {prime} -1} { operatorname {rad} ( text {prime} -1)} $ equals the sequence of primors

We denote whole numbers $ m> 1 $ share the product of different primes $ m $ as $$ operatorname {rad} (m) = prod_ { substack {p mid m \ p text {prime}}} p, $$
with the definition $ operatorname {rad} (1) = 1 $ (see you want the Wikipedia Radical of an integer), this is the famous multiplicative function in the statement of the abc conjecture. We also call that $ k $th primorial as $$ N_k = prod_ {t = 1} ^ k p_t $$ this is the product of the first $ k $ Primes.

I was inspired in the first paragraph of the section B46 (1) to propose a variant of the type of problem the book presents. I think about solutions $ (p, k) $, from where $ p $ always denotes a prime number for which the identity
$$ frac {p-1} { operatorname {rad} (p-1)} = N_k tag {1} $$
applies to some primitive ones $ N_k $With $ k geq 1 $,

I've used a Pari / GP program to write the first satisfying primes $ (1) $, these first few primes $ p $are (this is a selection of my calculations) $ 5,13,29,37,53,61,149,157,173,181,229, 269,277, $ $ 293,317, ldots $ which correspond to these indices $ k $& # 39; s
$ 1,1,1,2,1,1,1,1,1,2,1,1,1,1 ldots $ our origins $ N_k $ fulfill the equation $ (1) $,

The curiosity that I have is something about the size or cardinality of the set $$ mathcal {K} = {k geq 1: (p, k) text {is a solution to the equation} (1) }. $$

Guess. The sentence $ mathcal {K} $ is finally,

So what I'm saying is the set $ mathcal {K} $ remains as a finite / finite set of positive integers, if the other variable $ p $ Runs over the set of all primes.

Question. Can one refute earlier assumptions? Can you do some thinking or calculations
clarify the truthfulness of the conjecture? Many thanks.

Maybe they can be useful state boundaries or an approximation to cardinality $ | mathcal {K} | $ for increasing segments of primes $ 2 leq p leq X $ and primors with indices $ 1 leq k leq Y $

Under these circumstances, I have no idea whether previous guesses are true, but I said I just want to find solutions to the integers $ k = $ 1.2 and $ 3 $ (I do not know if my contribution to this sequence has good mathematical content, the comments are welcome).


(1) Richard K. Guy, Unresolved Problems in Number Theory, Unresolved Problems in Intuitive Mathematics Volume I, 2nd Edition, Springer-Verlag (1994).

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Given a value that I need to know until the number equals a list of numbers and their index

I have a list of ascending order of numbers. For a given value, I have to know until the number matches the array and its index.

list myLista = new list(New[] {"2", "9", "17", "25", "35", "42", "70"});

Initial index = 0.
At a value of 5, the index would be 1 and the number 9
At a value of 17, the index would be 2 and the number 17
For a value = 0, the index would be 0 and the number 2

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dom – XSS with escape equals sign in the jQuery selector

The site uses jQuery 1.8.3, which has a known XSS vulnerability in the selector. (Https://

It happens filtered and urdecoded ducument.location.hash (val2 below) value within the voter.

$ (& Div; div[data-foo=''+filter(val1)+''][data-value=''+filter(val2)+'']& # 39)

Function filter (str) {
if (str)
return str.replace (/ ([ #;?%&,.+*~':"!^$[]() => |  / @]) / g, & # 39; \ $ 1 & # 39;
return str;

I have reached the following payload:

It would work if the equals sign were not replaced by =, Browsers do not seem to tolerate = at all.

Any ideas how this can be bypassed? Or maybe another payload would work here?

Why do I get "java.lang.NullPointerException" each time I use equals ()? Java Poo

Well, I'm doing Poo programs and every time I want to use the equals () method, when I run the code, I jump red: "java.lang.NullPointerException", why is it going to be like this?

public class user {

private string user;
private string password;

public user (string name, string password) {

this.username = name;
this.password = password;

public String getName () {

return this.user;

public void setName (stringname) {

this.username = name;

public String getPassword () {

Please return this password.

public void setPassword (String password) {

this.password = password;


public class sistema {

private string name;
private user list[];
int usersadded = 0;

public system (string name, int number of users) { = name;
this.listDeUsers = new user[cantidadDeUsuarios];


public String getName () {

return this name;


public void setName (stringname) { = name;

public user[] getListaDeAutos (user list of users) {

returns this list of users;

public void setUserList (User[] User list) {

this.list of users = list of users;


public boolean loguearUser (string user, string password) {

Boolean state = false;
int i = 0;

for (i = 0; i <userlist.length; i ++) {
if (list of users[i].getName (). equals (user)
&& user list[i].getPassword (). equals (password)) {

State = true;


State = wrong;

Returning state;

public void addUsers (user user) {

Member list[usuariosAgregados] = User;
User aggregates ++;