Show that there exists infinitely many primes which satisfy a given congurence.
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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
asked Nov 18 at 10:52
Mittal G
1,143515
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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
asked Nov 18 at 10:52
Mittal G
1,143515
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?
– Samurai
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
asked Nov 18 at 10:52
Mittal G
1,143515
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
asked Nov 18 at 10:52
Mittal G
1,143515
asked Nov 18 at 10:52
Mittal G
1,143515
edited Nov 18 at 11:02
asked Nov 18 at 10:52
Mittal G
1,143515
asked Nov 18 at 10:52
Mittal G
1,143515
asked Nov 18 at 10:52
Mittal G
1,143515
1,143515
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?
– Samurai
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?
– Samurai
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?
– Samurai
Nov 20 at 13:29
Where did you find this problem?
– Samurai
Nov 20 at 13:29
add a comment |
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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?
– Samurai
Nov 20 at 13:29