Mixing
Confusion over proof of Lifting the Exponent Lemma
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Lifting the Exponent Lemma
I am referring to page 3 of the linked paper. In the paper, the author proves that $v_p(x^p-y^p) = v_p(x-y) +1$. To prove this, he first proves that p divides $fracx^p-y^px-p$. The proof for this makes sense to me although the reason behind his next step is quite unclear to me. For some reason, he proves that $p^2$ does not divide $fracx^p-y^px-p$. What is the purpose of doing this? Isn't it obvious that $p^2$ can't divide $px^p-1$ if $p$ can't divide $x$?
number-theory elementary-number-theory modular-arithmetic
edited Jul 31 at 0:29
Stefan4024
27.8k52974
asked Jul 30 at 23:17
Dude156
17612
add a comment |Â
up vote
1
down vote
favorite
Lifting the Exponent Lemma
I am referring to page 3 of the linked paper. In the paper, the author proves that $v_p(x^p-y^p) = v_p(x-y) +1$. To prove this, he first proves that p divides $fracx^p-y^px-p$. The proof for this makes sense to me although the reason behind his next step is quite unclear to me. For some reason, he proves that $p^2$ does not divide $fracx^p-y^px-p$. What is the purpose of doing this? Isn't it obvious that $p^2$ can't divide $px^p-1$ if $p$ can't divide $x$?
number-theory elementary-number-theory modular-arithmetic
edited Jul 31 at 0:29
Stefan4024
27.8k52974
asked Jul 30 at 23:17
Dude156
17612
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Lifting the Exponent Lemma
I am referring to page 3 of the linked paper. In the paper, the author proves that $v_p(x^p-y^p) = v_p(x-y) +1$. To prove this, he first proves that p divides $fracx^p-y^px-p$. The proof for this makes sense to me although the reason behind his next step is quite unclear to me. For some reason, he proves that $p^2$ does not divide $fracx^p-y^px-p$. What is the purpose of doing this? Isn't it obvious that $p^2$ can't divide $px^p-1$ if $p$ can't divide $x$?
number-theory elementary-number-theory modular-arithmetic
edited Jul 31 at 0:29
Stefan4024
27.8k52974
asked Jul 30 at 23:17
Dude156
17612
Lifting the Exponent Lemma
I am referring to page 3 of the linked paper. In the paper, the author proves that $v_p(x^p-y^p) = v_p(x-y) +1$. To prove this, he first proves that p divides $fracx^p-y^px-p$. The proof for this makes sense to me although the reason behind his next step is quite unclear to me. For some reason, he proves that $p^2$ does not divide $fracx^p-y^px-p$. What is the purpose of doing this? Isn't it obvious that $p^2$ can't divide $px^p-1$ if $p$ can't divide $x$?
number-theory elementary-number-theory modular-arithmetic
edited Jul 31 at 0:29
Stefan4024
27.8k52974
asked Jul 30 at 23:17
Dude156
17612
edited Jul 31 at 0:29
Stefan4024
27.8k52974
edited Jul 31 at 0:29
Stefan4024
27.8k52974
edited Jul 31 at 0:29
Stefan4024
27.8k52974
27.8k52974
asked Jul 30 at 23:17
Dude156
17612
asked Jul 30 at 23:17
Dude156
17612
asked Jul 30 at 23:17
Dude156
17612
17612
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
The reason why he proves that $p^2$ doesn't divide $fracx^p-y^px-y$ is to prove that $p$ is the highest power of $p$ dividing the integer.
I feel that the confusion stems from one of the first line in the proof where we have that $x^p-1y + cdots xy^p-1 equiv px^p-1 pmod p$. Note that here we have $p$ as a modulo and not $p^2$. Thus this isn't enough to conclude that the number on the left is equal to $px^p-1$ modulo $p^2$. Therefore we need the extra work.
answered Jul 31 at 0:35
Stefan4024
27.8k52974
aaaah i see! Thanks!
– Dude156
Jul 31 at 1:59
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
The reason why he proves that $p^2$ doesn't divide $fracx^p-y^px-y$ is to prove that $p$ is the highest power of $p$ dividing the integer.
I feel that the confusion stems from one of the first line in the proof where we have that $x^p-1y + cdots xy^p-1 equiv px^p-1 pmod p$. Note that here we have $p$ as a modulo and not $p^2$. Thus this isn't enough to conclude that the number on the left is equal to $px^p-1$ modulo $p^2$. Therefore we need the extra work.
answered Jul 31 at 0:35
Stefan4024
27.8k52974
aaaah i see! Thanks!
– Dude156
Jul 31 at 1:59
add a comment |Â
up vote
3
down vote
accepted
The reason why he proves that $p^2$ doesn't divide $fracx^p-y^px-y$ is to prove that $p$ is the highest power of $p$ dividing the integer.
I feel that the confusion stems from one of the first line in the proof where we have that $x^p-1y + cdots xy^p-1 equiv px^p-1 pmod p$. Note that here we have $p$ as a modulo and not $p^2$. Thus this isn't enough to conclude that the number on the left is equal to $px^p-1$ modulo $p^2$. Therefore we need the extra work.
answered Jul 31 at 0:35
Stefan4024
27.8k52974
aaaah i see! Thanks!
– Dude156
Jul 31 at 1:59
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
The reason why he proves that $p^2$ doesn't divide $fracx^p-y^px-y$ is to prove that $p$ is the highest power of $p$ dividing the integer.
I feel that the confusion stems from one of the first line in the proof where we have that $x^p-1y + cdots xy^p-1 equiv px^p-1 pmod p$. Note that here we have $p$ as a modulo and not $p^2$. Thus this isn't enough to conclude that the number on the left is equal to $px^p-1$ modulo $p^2$. Therefore we need the extra work.
answered Jul 31 at 0:35
Stefan4024
27.8k52974
The reason why he proves that $p^2$ doesn't divide $fracx^p-y^px-y$ is to prove that $p$ is the highest power of $p$ dividing the integer.
I feel that the confusion stems from one of the first line in the proof where we have that $x^p-1y + cdots xy^p-1 equiv px^p-1 pmod p$. Note that here we have $p$ as a modulo and not $p^2$. Thus this isn't enough to conclude that the number on the left is equal to $px^p-1$ modulo $p^2$. Therefore we need the extra work.
answered Jul 31 at 0:35
Stefan4024
27.8k52974
answered Jul 31 at 0:35
Stefan4024
27.8k52974
answered Jul 31 at 0:35
Stefan4024
27.8k52974
answered Jul 31 at 0:35
Stefan4024
27.8k52974
27.8k52974
aaaah i see! Thanks!
– Dude156
Jul 31 at 1:59
add a comment |Â
aaaah i see! Thanks!
– Dude156
Jul 31 at 1:59
aaaah i see! Thanks!
– Dude156
Jul 31 at 1:59
aaaah i see! Thanks!
– Dude156
Jul 31 at 1:59
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%2f2867505%2fconfusion-over-proof-of-lifting-the-exponent-lemma%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