Mixing
Sums of two integer squares in arithmetic progressions
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Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to count integer representations of $n$ by the quadratic form $x^2+ 5^2 y^2$.
I would hope for something as nice as the formula related to representations as a sum of two integer squares,
$$ sum_{substack{ din mathbb{N} \ d text{ divides } n}} chi(d),$$ where $chi$ is the non-principal character modulo $4$.
In general, I would be interested in the number of representations by any quadratic form of the shape $d_1^2 x^2+ d_2^2 y^2$, where $d_1,d_2$ are non-zero integers.
nt.number-theory algebraic-number-theory quadratic-forms
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
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add a comment |
$begingroup$
Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to count integer representations of $n$ by the quadratic form $x^2+ 5^2 y^2$.
I would hope for something as nice as the formula related to representations as a sum of two integer squares,
$$ sum_{substack{ din mathbb{N} \ d text{ divides } n}} chi(d),$$ where $chi$ is the non-principal character modulo $4$.
In general, I would be interested in the number of representations by any quadratic form of the shape $d_1^2 x^2+ d_2^2 y^2$, where $d_1,d_2$ are non-zero integers.
nt.number-theory algebraic-number-theory quadratic-forms
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
$endgroup$
4
$begingroup$
Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180
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– GH from MO
Feb 1 at 20:14
$begingroup$
yes that is a great but how is it useful? the square coefficients make all the quadratic forms in the question equivalent to $x^2+y^2$ so not sure how quadratic fields etc.etc. are relevant.
$endgroup$
– Captain Darling
Feb 7 at 8:33
$begingroup$
I believe that the kind of formula you expect does not exist. Unfortuately, I am no expert here, and I don't have the time to check the literature.
$endgroup$
– GH from MO
Feb 7 at 18:50
add a comment |
$begingroup$
Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to count integer representations of $n$ by the quadratic form $x^2+ 5^2 y^2$.
I would hope for something as nice as the formula related to representations as a sum of two integer squares,
$$ sum_{substack{ din mathbb{N} \ d text{ divides } n}} chi(d),$$ where $chi$ is the non-principal character modulo $4$.
In general, I would be interested in the number of representations by any quadratic form of the shape $d_1^2 x^2+ d_2^2 y^2$, where $d_1,d_2$ are non-zero integers.
nt.number-theory algebraic-number-theory quadratic-forms
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
$endgroup$
Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to count integer representations of $n$ by the quadratic form $x^2+ 5^2 y^2$.
I would hope for something as nice as the formula related to representations as a sum of two integer squares,
$$ sum_{substack{ din mathbb{N} \ d text{ divides } n}} chi(d),$$ where $chi$ is the non-principal character modulo $4$.
In general, I would be interested in the number of representations by any quadratic form of the shape $d_1^2 x^2+ d_2^2 y^2$, where $d_1,d_2$ are non-zero integers.
nt.number-theory algebraic-number-theory quadratic-forms
nt.number-theory algebraic-number-theory quadratic-forms
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
asked Feb 1 at 15:59
Captain DarlingCaptain Darling
1,251818
1,251818
4
$begingroup$
Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180
$endgroup$
– GH from MO
Feb 1 at 20:14
$begingroup$
yes that is a great but how is it useful? the square coefficients make all the quadratic forms in the question equivalent to $x^2+y^2$ so not sure how quadratic fields etc.etc. are relevant.
$endgroup$
– Captain Darling
Feb 7 at 8:33
$begingroup$
I believe that the kind of formula you expect does not exist. Unfortuately, I am no expert here, and I don't have the time to check the literature.
$endgroup$
– GH from MO
Feb 7 at 18:50
add a comment |
4
$begingroup$
Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180
$endgroup$
– GH from MO
Feb 1 at 20:14
$begingroup$
yes that is a great but how is it useful? the square coefficients make all the quadratic forms in the question equivalent to $x^2+y^2$ so not sure how quadratic fields etc.etc. are relevant.
$endgroup$
– Captain Darling
Feb 7 at 8:33
$begingroup$
I believe that the kind of formula you expect does not exist. Unfortuately, I am no expert here, and I don't have the time to check the literature.
$endgroup$
– GH from MO
Feb 7 at 18:50
4
4
$begingroup$
Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180
$endgroup$
– GH from MO
Feb 1 at 20:14
$begingroup$
Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180
$endgroup$
– GH from MO
Feb 1 at 20:14
$begingroup$
yes that is a great but how is it useful? the square coefficients make all the quadratic forms in the question equivalent to $x^2+y^2$ so not sure how quadratic fields etc.etc. are relevant.
$endgroup$
– Captain Darling
Feb 7 at 8:33
$begingroup$
yes that is a great but how is it useful? the square coefficients make all the quadratic forms in the question equivalent to $x^2+y^2$ so not sure how quadratic fields etc.etc. are relevant.
$endgroup$
– Captain Darling
Feb 7 at 8:33
$begingroup$
I believe that the kind of formula you expect does not exist. Unfortuately, I am no expert here, and I don't have the time to check the literature.
$endgroup$
– GH from MO
Feb 7 at 18:50
$begingroup$
I believe that the kind of formula you expect does not exist. Unfortuately, I am no expert here, and I don't have the time to check the literature.
$endgroup$
– GH from MO
Feb 7 at 18:50
add a comment |
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$begingroup$
Things are not so simple in general. Have a look at the book amazon.com/Primes-Form-x2-ny2-Multiplication/dp/1118390180
$endgroup$
– GH from MO
Feb 1 at 20:14
$begingroup$
yes that is a great but how is it useful? the square coefficients make all the quadratic forms in the question equivalent to $x^2+y^2$ so not sure how quadratic fields etc.etc. are relevant.
$endgroup$
– Captain Darling
Feb 7 at 8:33
$begingroup$
I believe that the kind of formula you expect does not exist. Unfortuately, I am no expert here, and I don't have the time to check the literature.
$endgroup$
– GH from MO
Feb 7 at 18:50