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It's known that the number of representations of an integer $k$ by sum of two squares is $$ 4\;\sum_{d|k}\left(\frac{-4}{d}\right) $$ or $$ 4\sum_{d|k,\; d \textrm{ odd}} (-1)^{\frac{d-1}{2}}= 4(d_1(k)-d_3(k)) $$ where $d_1(k)$ and $d_3(k)$ are the numbers of the divisors of $k$ of the forms $4m+1$ or $4m+3$ respectively.

I would like to have references about similar formulas for the number of representations of an integer by the quadratic form $$ x^2+N\;y^2 $$ with $N$ a positive integer, or more generally, any positive definite binary quadratic form. Thanks.-.

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You might find this other mathoverflow question useful: mathoverflow.net/questions/79342/… –  Daniel Loughran Jan 15 '12 at 21:03

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up vote 7 down vote accepted

There are going to be very few of these that come out so cleanly. There is not going to be anything clean unless there is only one class per genus "idoneal" (your discriminant $-4N$), and I would not really be confident unless there is only one class entirely for that discriminant. In the latter case, we have your $N=1,2,3,4,7.$

For possibilities with more general types of expressions, see BERKOVICH

Here we go, Leonard Eugene Dickson, Introduction to the Theory of Numbers (1929), exercises on pages 80-81, for section 51. If $m$ is positive and odd, the number of representations of $2^k m$ by $x^2 + 2 y^2$ is double the excess of the number of divisors $\equiv 1 \; \mbox{or} \; 3 \pmod 8$ of $m$ over the number of divisors $\equiv 5 \; \mbox{or} \; 7 \pmod 8$ of $m.$

The number of representations of any positive $n$ by $x^2 + x y + y^2$ is $6E(n),$ where $E(n)$ is excess of the number of divisors $\equiv 1 \pmod 3$ of $n$ over the number of divisors $\equiv 2 \pmod 3$ of $n.$ Also, if $n=2^k m$ with $m$ odd, then $E(n)=0$ when $k$ is odd, while $E(n) = E(m)$ when $k$ is even.

Using the same $E(n)$ as above, the number of representations of $2^k m$ with $m$ odd by $x^2 + 3 y^2$ is $0$ if $k$ is odd, $2E(m)$ if $k=0,$ and $6 E(m) $ if $k$ is even and nonzero.

Dickson sticks with one class per discriminant on pages 80-81, then allows one class per genus on pages 84-86.

Note that we cannot expect such simple formulas when there are more than one classes per genus. If there were pretty formulas, we would have a simple way to decide whether an individual prime were represented. However, for example, a prime $p \equiv 1 \pmod 3$ is represented by $x^2 + 27 y^2$ if and only if 2 is a cubic residue $\pmod p.$ If $p \neq 7,2$ and $(-14|p) = 1,$ then $p = u^2 + 14 v^2$ is possible in integers if and only if there is an integer solution to $(z^2 + 1)^2 \equiv 8 \pmod p.$

Here is a good one. If $m$ is positive and odd, the number of representations of $m$ by $x^2 + xy + 3 y^2$ is double the excess of the number of divisors $\equiv 1, \; 3, \; 4, \; 5, \; \mbox{or} \; 9 \pmod {11}$ of $m$ over the number of divisors $\equiv 2, \; 6, \; 7, \; 8, \; \mbox{or} \; 10 \pmod {11}$ of $m.$ Now, a subset of these numbers is represented by $x^2 + 11 y^2,$ by a simple formula. However, for a single prime, if $p > 11$ has $ (-44 | p) = 1,$ then we have an integral representation $p = u^2 + 11 v^2$ if and only if $z^3 + z^2 - z + 1 \equiv 0 \pmod p$ has an integral solution. But there is always a representation $ p = x^2 + x y + 3 y^2$ as soon as $ (-44 | p) = 1.$

In case of interest, the primes up to 500 represented by $x^2 + 11 y^2$ are

11 47 53 103 163 199 257 269 311 397 401 419 421 499

while the primes up to 500 represented by $3 x^2 \pm 2 x y + 4 y^2$ are

3 5 23 31 37 59 67 71 89 97 113 137 157 179 181 191 223 229 251 313 317 331 353 367 379 383 389 433 443 449 463 467 487

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