Mixing
Counting Lattice Points
1
$begingroup$
Lattice points are of great importance. I encountered a problem as follows:- given a circle of radius 'r' as x²+y²=r² the number of lattice points can take values . The options were (0,72,69,140). I tried the problem using complex numbers but couldn't figure out a short and sweet method to tackle the problem so I need help ! Please help me!😊
elementary-number-theory complex-numbers integer-lattices
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
$endgroup$
add a comment |
1
$begingroup$
Lattice points are of great importance. I encountered a problem as follows:- given a circle of radius 'r' as x²+y²=r² the number of lattice points can take values . The options were (0,72,69,140). I tried the problem using complex numbers but couldn't figure out a short and sweet method to tackle the problem so I need help ! Please help me!😊
elementary-number-theory complex-numbers integer-lattices
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
$endgroup$
$begingroup$
Are you counting the lattice points inside the circle, or the the lattice points on the circle itself?
$endgroup$
– Dylan
Feb 1 at 10:35
add a comment |
1
1
1
$begingroup$
Lattice points are of great importance. I encountered a problem as follows:- given a circle of radius 'r' as x²+y²=r² the number of lattice points can take values . The options were (0,72,69,140). I tried the problem using complex numbers but couldn't figure out a short and sweet method to tackle the problem so I need help ! Please help me!😊
elementary-number-theory complex-numbers integer-lattices
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
$endgroup$
Lattice points are of great importance. I encountered a problem as follows:- given a circle of radius 'r' as x²+y²=r² the number of lattice points can take values . The options were (0,72,69,140). I tried the problem using complex numbers but couldn't figure out a short and sweet method to tackle the problem so I need help ! Please help me!😊
elementary-number-theory complex-numbers integer-lattices
elementary-number-theory complex-numbers integer-lattices
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
edited Jan 31 at 20:18
Jyrki Lahtonen
110k13172390
110k13172390
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
asked Jan 31 at 19:02
Aditya GargAditya Garg
23413
23413
$begingroup$
Are you counting the lattice points inside the circle, or the the lattice points on the circle itself?
$endgroup$
– Dylan
Feb 1 at 10:35
add a comment |
$begingroup$
Are you counting the lattice points inside the circle, or the the lattice points on the circle itself?
$endgroup$
– Dylan
Feb 1 at 10:35
$begingroup$
Are you counting the lattice points inside the circle, or the the lattice points on the circle itself?
$endgroup$
– Dylan
Feb 1 at 10:35
$begingroup$
Are you counting the lattice points inside the circle, or the the lattice points on the circle itself?
$endgroup$
– Dylan
Feb 1 at 10:35
add a comment |
1 Answer
1
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5
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Hint, if you believe that exactly one of the 4 given choices is correct: the center of the circle is a lattice point, and has a property all the other lattice points lack.
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
$endgroup$
$begingroup$
A great hint. Even though we (ideally) stll need to perform a reality check given that there are two different sizes for the non-trivial orbits of $D_4$.
$endgroup$
– Jyrki Lahtonen
Jan 31 at 20:20
$begingroup$
Can I do something like n=(complex prime factorisation) and then use some combinatorics to end up at 4(pi(sigma{X(i^r)})) and thanks for the hint
$endgroup$
– Aditya Garg
Jan 31 at 20:33
$begingroup$
Do you mean use Gauss integers factorization ?
$endgroup$
– Jean Marie
Jan 31 at 21:12
$begingroup$
Do we agree that your lattice is $mathbb{Z} times mathbb{Z}$ with center at the origin ?
$endgroup$
– Jean Marie
Jan 31 at 21:17
2
$begingroup$
Yes. The number of lattice points inside the circle must be odd: the origin plus twice the number of pairs of non-zero lattice points $pmlambda$.
$endgroup$
– kimchi lover
Jan 31 at 21:23
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
5
$begingroup$
Hint, if you believe that exactly one of the 4 given choices is correct: the center of the circle is a lattice point, and has a property all the other lattice points lack.
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
$endgroup$
$begingroup$
A great hint. Even though we (ideally) stll need to perform a reality check given that there are two different sizes for the non-trivial orbits of $D_4$.
$endgroup$
– Jyrki Lahtonen
Jan 31 at 20:20
$begingroup$
Can I do something like n=(complex prime factorisation) and then use some combinatorics to end up at 4(pi(sigma{X(i^r)})) and thanks for the hint
$endgroup$
– Aditya Garg
Jan 31 at 20:33
$begingroup$
Do you mean use Gauss integers factorization ?
$endgroup$
– Jean Marie
Jan 31 at 21:12
$begingroup$
Do we agree that your lattice is $mathbb{Z} times mathbb{Z}$ with center at the origin ?
$endgroup$
– Jean Marie
Jan 31 at 21:17
2
$begingroup$
Yes. The number of lattice points inside the circle must be odd: the origin plus twice the number of pairs of non-zero lattice points $pmlambda$.
$endgroup$
– kimchi lover
Jan 31 at 21:23
add a comment |
5
$begingroup$
Hint, if you believe that exactly one of the 4 given choices is correct: the center of the circle is a lattice point, and has a property all the other lattice points lack.
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
$endgroup$
$begingroup$
A great hint. Even though we (ideally) stll need to perform a reality check given that there are two different sizes for the non-trivial orbits of $D_4$.
$endgroup$
– Jyrki Lahtonen
Jan 31 at 20:20
$begingroup$
Can I do something like n=(complex prime factorisation) and then use some combinatorics to end up at 4(pi(sigma{X(i^r)})) and thanks for the hint
$endgroup$
– Aditya Garg
Jan 31 at 20:33
$begingroup$
Do you mean use Gauss integers factorization ?
$endgroup$
– Jean Marie
Jan 31 at 21:12
$begingroup$
Do we agree that your lattice is $mathbb{Z} times mathbb{Z}$ with center at the origin ?
$endgroup$
– Jean Marie
Jan 31 at 21:17
2
$begingroup$
Yes. The number of lattice points inside the circle must be odd: the origin plus twice the number of pairs of non-zero lattice points $pmlambda$.
$endgroup$
– kimchi lover
Jan 31 at 21:23
add a comment |
5
5
5
$begingroup$
Hint, if you believe that exactly one of the 4 given choices is correct: the center of the circle is a lattice point, and has a property all the other lattice points lack.
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
$endgroup$
Hint, if you believe that exactly one of the 4 given choices is correct: the center of the circle is a lattice point, and has a property all the other lattice points lack.
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
answered Jan 31 at 20:00
kimchi loverkimchi lover
11.7k31229
11.7k31229
$begingroup$
A great hint. Even though we (ideally) stll need to perform a reality check given that there are two different sizes for the non-trivial orbits of $D_4$.
$endgroup$
– Jyrki Lahtonen
Jan 31 at 20:20
$begingroup$
Can I do something like n=(complex prime factorisation) and then use some combinatorics to end up at 4(pi(sigma{X(i^r)})) and thanks for the hint
$endgroup$
– Aditya Garg
Jan 31 at 20:33
$begingroup$
Do you mean use Gauss integers factorization ?
$endgroup$
– Jean Marie
Jan 31 at 21:12
$begingroup$
Do we agree that your lattice is $mathbb{Z} times mathbb{Z}$ with center at the origin ?
$endgroup$
– Jean Marie
Jan 31 at 21:17
2
$begingroup$
Yes. The number of lattice points inside the circle must be odd: the origin plus twice the number of pairs of non-zero lattice points $pmlambda$.
$endgroup$
– kimchi lover
Jan 31 at 21:23
add a comment |
$begingroup$
A great hint. Even though we (ideally) stll need to perform a reality check given that there are two different sizes for the non-trivial orbits of $D_4$.
$endgroup$
– Jyrki Lahtonen
Jan 31 at 20:20
$begingroup$
Can I do something like n=(complex prime factorisation) and then use some combinatorics to end up at 4(pi(sigma{X(i^r)})) and thanks for the hint
$endgroup$
– Aditya Garg
Jan 31 at 20:33
$begingroup$
Do you mean use Gauss integers factorization ?
$endgroup$
– Jean Marie
Jan 31 at 21:12
$begingroup$
Do we agree that your lattice is $mathbb{Z} times mathbb{Z}$ with center at the origin ?
$endgroup$
– Jean Marie
Jan 31 at 21:17
2
$begingroup$
Yes. The number of lattice points inside the circle must be odd: the origin plus twice the number of pairs of non-zero lattice points $pmlambda$.
$endgroup$
– kimchi lover
Jan 31 at 21:23
$begingroup$
A great hint. Even though we (ideally) stll need to perform a reality check given that there are two different sizes for the non-trivial orbits of $D_4$.
$endgroup$
– Jyrki Lahtonen
Jan 31 at 20:20
$begingroup$
A great hint. Even though we (ideally) stll need to perform a reality check given that there are two different sizes for the non-trivial orbits of $D_4$.
$endgroup$
– Jyrki Lahtonen
Jan 31 at 20:20
$begingroup$
Can I do something like n=(complex prime factorisation) and then use some combinatorics to end up at 4(pi(sigma{X(i^r)})) and thanks for the hint
$endgroup$
– Aditya Garg
Jan 31 at 20:33
$begingroup$
Can I do something like n=(complex prime factorisation) and then use some combinatorics to end up at 4(pi(sigma{X(i^r)})) and thanks for the hint
$endgroup$
– Aditya Garg
Jan 31 at 20:33
$begingroup$
Do you mean use Gauss integers factorization ?
$endgroup$
– Jean Marie
Jan 31 at 21:12
$begingroup$
Do you mean use Gauss integers factorization ?
$endgroup$
– Jean Marie
Jan 31 at 21:12
$begingroup$
Do we agree that your lattice is $mathbb{Z} times mathbb{Z}$ with center at the origin ?
$endgroup$
– Jean Marie
Jan 31 at 21:17
$begingroup$
Do we agree that your lattice is $mathbb{Z} times mathbb{Z}$ with center at the origin ?
$endgroup$
– Jean Marie
Jan 31 at 21:17
2
2
$begingroup$
Yes. The number of lattice points inside the circle must be odd: the origin plus twice the number of pairs of non-zero lattice points $pmlambda$.
$endgroup$
– kimchi lover
Jan 31 at 21:23
$begingroup$
Yes. The number of lattice points inside the circle must be odd: the origin plus twice the number of pairs of non-zero lattice points $pmlambda$.
$endgroup$
– kimchi lover
Jan 31 at 21:23
add a comment |
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$begingroup$
Are you counting the lattice points inside the circle, or the the lattice points on the circle itself?
$endgroup$
– Dylan
Feb 1 at 10:35