Solutions to the equation $3m^2equiv 1pmodp$ where $p$ is some prime
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Here is my question:
Solve the equation $3m^2equiv 1pmodp$, where $p$ is some prime.
This seems to be closely related to quadratic residues. For example, if we were to solve $x^2equiv -1pmodp$, we can conclude that the equation has solutions if and only if $pequiv 1pmod4$, using the famous Euler's criterion,
$$left(frac-1pright)equiv (-1)^fracp-12pmodp.$$
However, since now the square is multiplied by $3$, I had no idea what to do. I tried multiplying the inverse of $3$ modulo $p$ (assume $pneq 3$), to get $m^2equiv 3^-1pmodp$ but I don't know if I did anything wrong or is this even the right path to go.
Can anyone help? Thanks in advance.
quadratic-residues
asked Aug 15 at 6:48
blastzit
344110
add a comment |Â
up vote
0
down vote
favorite
Here is my question:
Solve the equation $3m^2equiv 1pmodp$, where $p$ is some prime.
This seems to be closely related to quadratic residues. For example, if we were to solve $x^2equiv -1pmodp$, we can conclude that the equation has solutions if and only if $pequiv 1pmod4$, using the famous Euler's criterion,
$$left(frac-1pright)equiv (-1)^fracp-12pmodp.$$
However, since now the square is multiplied by $3$, I had no idea what to do. I tried multiplying the inverse of $3$ modulo $p$ (assume $pneq 3$), to get $m^2equiv 3^-1pmodp$ but I don't know if I did anything wrong or is this even the right path to go.
Can anyone help? Thanks in advance.
quadratic-residues
asked Aug 15 at 6:48
blastzit
344110
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Here is my question:
Solve the equation $3m^2equiv 1pmodp$, where $p$ is some prime.
This seems to be closely related to quadratic residues. For example, if we were to solve $x^2equiv -1pmodp$, we can conclude that the equation has solutions if and only if $pequiv 1pmod4$, using the famous Euler's criterion,
$$left(frac-1pright)equiv (-1)^fracp-12pmodp.$$
However, since now the square is multiplied by $3$, I had no idea what to do. I tried multiplying the inverse of $3$ modulo $p$ (assume $pneq 3$), to get $m^2equiv 3^-1pmodp$ but I don't know if I did anything wrong or is this even the right path to go.
Can anyone help? Thanks in advance.
quadratic-residues
asked Aug 15 at 6:48
blastzit
344110
Here is my question:
Solve the equation $3m^2equiv 1pmodp$, where $p$ is some prime.
This seems to be closely related to quadratic residues. For example, if we were to solve $x^2equiv -1pmodp$, we can conclude that the equation has solutions if and only if $pequiv 1pmod4$, using the famous Euler's criterion,
$$left(frac-1pright)equiv (-1)^fracp-12pmodp.$$
However, since now the square is multiplied by $3$, I had no idea what to do. I tried multiplying the inverse of $3$ modulo $p$ (assume $pneq 3$), to get $m^2equiv 3^-1pmodp$ but I don't know if I did anything wrong or is this even the right path to go.
Can anyone help? Thanks in advance.
quadratic-residues
asked Aug 15 at 6:48
blastzit
344110
asked Aug 15 at 6:48
blastzit
344110
asked Aug 15 at 6:48
blastzit
344110
asked Aug 15 at 6:48
blastzit
344110
344110
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Multiply through by 3, getting $(3m)^2equiv3modp$.
Then there are clearly only solutions if 3 is a quadratic residue. Suppose it is, and $3equiva^2$. Then $3mequivpma$.
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Multiply through by 3, getting $(3m)^2equiv3modp$.
Then there are clearly only solutions if 3 is a quadratic residue. Suppose it is, and $3equiva^2$. Then $3mequivpma$.
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
add a comment |Â
up vote
1
down vote
accepted
Multiply through by 3, getting $(3m)^2equiv3modp$.
Then there are clearly only solutions if 3 is a quadratic residue. Suppose it is, and $3equiva^2$. Then $3mequivpma$.
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Multiply through by 3, getting $(3m)^2equiv3modp$.
Then there are clearly only solutions if 3 is a quadratic residue. Suppose it is, and $3equiva^2$. Then $3mequivpma$.
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
Multiply through by 3, getting $(3m)^2equiv3modp$.
Then there are clearly only solutions if 3 is a quadratic residue. Suppose it is, and $3equiva^2$. Then $3mequivpma$.
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
answered Aug 15 at 7:30
Benedict Randall Shaw
3139
3139
add a comment |Â
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2883263%2fsolutions-to-the-equation-3m2-equiv-1-pmodp-where-p-is-some-prime%23new-answer', 'question_page');
);
Post as a guest
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
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
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
