Are rational points dense on every circle in the coordinate plane?
Clash Royale CLAN TAG#URR8PPP
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1
down vote
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Are rational points dense on every circle in the coordinate plane?
First thing first I know that rational points are dense on the unit circle.
However, I am not so sure how to show that rational points are not dense on every circle.
How would one come about answering this.
Any hits are appreciate it.
coordinate-systems rational-numbers
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up vote
1
down vote
favorite
Are rational points dense on every circle in the coordinate plane?
First thing first I know that rational points are dense on the unit circle.
However, I am not so sure how to show that rational points are not dense on every circle.
How would one come about answering this.
Any hits are appreciate it.
coordinate-systems rational-numbers
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Are rational points dense on every circle in the coordinate plane?
First thing first I know that rational points are dense on the unit circle.
However, I am not so sure how to show that rational points are not dense on every circle.
How would one come about answering this.
Any hits are appreciate it.
coordinate-systems rational-numbers
Are rational points dense on every circle in the coordinate plane?
First thing first I know that rational points are dense on the unit circle.
However, I am not so sure how to show that rational points are not dense on every circle.
How would one come about answering this.
Any hits are appreciate it.
coordinate-systems rational-numbers
coordinate-systems rational-numbers
edited 2 hours ago
asked 2 hours ago
Hidaw
110219
110219
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2 Answers
2
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oldest
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up vote
2
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accepted
In particular, $x^2+y^2=3$ cannot have any rational points. If it had any such points then there would be integers $a,b,c$ having no common factor such that $(a/c)^2+(b/c)^2=3$ therefore $a^2+b^2=3c^2$. But with no common factor at least one of $a,b,c$ must be odd and all possibilities conforming with this requirement fail $bmod 4$.
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up vote
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They're not. No two different circles centered at the origin contain any of the same points. There are uncountably many circles, so most circles contain no rational points at all.
1
For example, the circle with center at the origin and radius $pi$ has no rational points. Replace $pi$ with any number whose square is irrational.
â GEdgar
2 hours ago
I had to read this six times before it hit me. Amazing.
â Randall
2 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
In particular, $x^2+y^2=3$ cannot have any rational points. If it had any such points then there would be integers $a,b,c$ having no common factor such that $(a/c)^2+(b/c)^2=3$ therefore $a^2+b^2=3c^2$. But with no common factor at least one of $a,b,c$ must be odd and all possibilities conforming with this requirement fail $bmod 4$.
add a comment |Â
up vote
2
down vote
accepted
In particular, $x^2+y^2=3$ cannot have any rational points. If it had any such points then there would be integers $a,b,c$ having no common factor such that $(a/c)^2+(b/c)^2=3$ therefore $a^2+b^2=3c^2$. But with no common factor at least one of $a,b,c$ must be odd and all possibilities conforming with this requirement fail $bmod 4$.
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
In particular, $x^2+y^2=3$ cannot have any rational points. If it had any such points then there would be integers $a,b,c$ having no common factor such that $(a/c)^2+(b/c)^2=3$ therefore $a^2+b^2=3c^2$. But with no common factor at least one of $a,b,c$ must be odd and all possibilities conforming with this requirement fail $bmod 4$.
In particular, $x^2+y^2=3$ cannot have any rational points. If it had any such points then there would be integers $a,b,c$ having no common factor such that $(a/c)^2+(b/c)^2=3$ therefore $a^2+b^2=3c^2$. But with no common factor at least one of $a,b,c$ must be odd and all possibilities conforming with this requirement fail $bmod 4$.
answered 38 mins ago
Oscar Lanzi
10.8k11734
10.8k11734
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up vote
5
down vote
They're not. No two different circles centered at the origin contain any of the same points. There are uncountably many circles, so most circles contain no rational points at all.
1
For example, the circle with center at the origin and radius $pi$ has no rational points. Replace $pi$ with any number whose square is irrational.
â GEdgar
2 hours ago
I had to read this six times before it hit me. Amazing.
â Randall
2 hours ago
add a comment |Â
up vote
5
down vote
They're not. No two different circles centered at the origin contain any of the same points. There are uncountably many circles, so most circles contain no rational points at all.
1
For example, the circle with center at the origin and radius $pi$ has no rational points. Replace $pi$ with any number whose square is irrational.
â GEdgar
2 hours ago
I had to read this six times before it hit me. Amazing.
â Randall
2 hours ago
add a comment |Â
up vote
5
down vote
up vote
5
down vote
They're not. No two different circles centered at the origin contain any of the same points. There are uncountably many circles, so most circles contain no rational points at all.
They're not. No two different circles centered at the origin contain any of the same points. There are uncountably many circles, so most circles contain no rational points at all.
answered 2 hours ago
Matt Samuel
35.2k43060
35.2k43060
1
For example, the circle with center at the origin and radius $pi$ has no rational points. Replace $pi$ with any number whose square is irrational.
â GEdgar
2 hours ago
I had to read this six times before it hit me. Amazing.
â Randall
2 hours ago
add a comment |Â
1
For example, the circle with center at the origin and radius $pi$ has no rational points. Replace $pi$ with any number whose square is irrational.
â GEdgar
2 hours ago
I had to read this six times before it hit me. Amazing.
â Randall
2 hours ago
1
1
For example, the circle with center at the origin and radius $pi$ has no rational points. Replace $pi$ with any number whose square is irrational.
â GEdgar
2 hours ago
For example, the circle with center at the origin and radius $pi$ has no rational points. Replace $pi$ with any number whose square is irrational.
â GEdgar
2 hours ago
I had to read this six times before it hit me. Amazing.
â Randall
2 hours ago
I had to read this six times before it hit me. Amazing.
â Randall
2 hours ago
add a comment |Â
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