Show that there exists infinitely many primes which satisfy a given congurence.
up vote
1
down vote
favorite
Let $m$ be a fixed positive integer that is the product of distinct prime factors of the form $(3k+2)$, such as $5 times 11$.
Prove that there exist infinitely many primes $p$ such that $3^{3p-2}equiv 1 pmod m$?
I started with assuming that there exists finitely many primes satisfying such equation. How to contradict this statement? May be we can create one more prime satisfying given equation? I am stucked. Please help
number-theory prime-numbers modular-arithmetic
add a comment |
up vote
1
down vote
favorite
Let $m$ be a fixed positive integer that is the product of distinct prime factors of the form $(3k+2)$, such as $5 times 11$.
Prove that there exist infinitely many primes $p$ such that $3^{3p-2}equiv 1 pmod m$?
I started with assuming that there exists finitely many primes satisfying such equation. How to contradict this statement? May be we can create one more prime satisfying given equation? I am stucked. Please help
number-theory prime-numbers modular-arithmetic
What is the order of $3 bmod 3k+2$ and $bmod m$ and how does it affect $3p-2$
– reuns
Nov 18 at 15:45
Where did you find this problem?
– user612946
Nov 20 at 13:29
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $m$ be a fixed positive integer that is the product of distinct prime factors of the form $(3k+2)$, such as $5 times 11$.
Prove that there exist infinitely many primes $p$ such that $3^{3p-2}equiv 1 pmod m$?
I started with assuming that there exists finitely many primes satisfying such equation. How to contradict this statement? May be we can create one more prime satisfying given equation? I am stucked. Please help
number-theory prime-numbers modular-arithmetic
Let $m$ be a fixed positive integer that is the product of distinct prime factors of the form $(3k+2)$, such as $5 times 11$.
Prove that there exist infinitely many primes $p$ such that $3^{3p-2}equiv 1 pmod m$?
I started with assuming that there exists finitely many primes satisfying such equation. How to contradict this statement? May be we can create one more prime satisfying given equation? I am stucked. Please help
number-theory prime-numbers modular-arithmetic
number-theory prime-numbers modular-arithmetic
edited Nov 18 at 11:02
asked Nov 18 at 10:52
Mittal G
1,180515
1,180515
What is the order of $3 bmod 3k+2$ and $bmod m$ and how does it affect $3p-2$
– reuns
Nov 18 at 15:45
Where did you find this problem?
– user612946
Nov 20 at 13:29
add a comment |
What is the order of $3 bmod 3k+2$ and $bmod m$ and how does it affect $3p-2$
– reuns
Nov 18 at 15:45
Where did you find this problem?
– user612946
Nov 20 at 13:29
What is the order of $3 bmod 3k+2$ and $bmod m$ and how does it affect $3p-2$
– reuns
Nov 18 at 15:45
What is the order of $3 bmod 3k+2$ and $bmod m$ and how does it affect $3p-2$
– reuns
Nov 18 at 15:45
Where did you find this problem?
– user612946
Nov 20 at 13:29
Where did you find this problem?
– user612946
Nov 20 at 13:29
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3003387%2fshow-that-there-exists-infinitely-many-primes-which-satisfy-a-given-congurence%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
What is the order of $3 bmod 3k+2$ and $bmod m$ and how does it affect $3p-2$
– reuns
Nov 18 at 15:45
Where did you find this problem?
– user612946
Nov 20 at 13:29