For which integers $D$ does $\left(\frac{-D}{p}\right)=1$ imply $p=x^2+Dy^2$?

Question

Let $D$ be an integer different from $0$ and $-1$ , such that $|D|$ is not a perfect square exceeding $1$.

A necessary condition for a prime number $p$ being expressible as $$p = x^2 + Dy^2$$ is $$\left(\frac{-D}{p}\right) = 1$$

For which integers $D$ is this condition also sufficient ?

For $|D| \le 2$ , infinite descent can be applied and possibly for $|D| = 3$ (who can clarify that ?)

Do the idoneal numbers play a role here ? Or do we need another approach , like the quadratic number fields ?

Additional question : For which numbers $D$ don't we have any possibility to classify the prime numbers of the form $x^2 + Dy^2$ ?

Answer 1

one thing you have not done is distinguish between discriminant $p$ (or $-p$) when $p$ is prime, compared with $-4p$ or $4p.$

The primes represented by $x^2 + xy + 3 y^2$ are entirely predictable. Discriminant is $-11,$ all you need to know.

However, the primes represented by $x^2 + 11 y^2$ are those $p$ for which $-11$ is a residue AND $$ x^3 + x^2 - x + 1 $$ factors $\pmod p$ as three distinct linear factors. The other primes are represented by $3 x^2 + 2 xy + 4 y^2.$

Discriminant $37:$ The primes represented by $x^2 + xy - 9 y^2$ are entirely predictable. Discriminant is $37,$ all you need to know.

However, the primes represented by $x^2 - 37 y^2$ are those $p$ for which $37$ is a residue AND a certain monic cubic $$ x^3 + x^2 - 3x - 1 $$ factors $\pmod p$ as three distinct linear factors. The other primes are represented by $3 x^2 + 2 xy -12 y^2.$

Here are the first few primes $x^2 - 37 y^2$

    37    67   107   137   139   151   233   269   293   317
   349   367   491   601   691   823   839   863   877   881
   929   941   971
parisize = 4000000, primelimit = 500000
? factormod( x^3 + x^2 - 3 * x - 1, 37)
%1 = 
[ Mod(1, 37)*x + Mod(4, 37) 2]

[Mod(1, 37)*x + Mod(30, 37) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 67)
%2 = 
[Mod(1, 67)*x + Mod(10, 67) 1]

[Mod(1, 67)*x + Mod(62, 67) 1]

[Mod(1, 67)*x + Mod(63, 67) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 107)
%3 = 
[Mod(1, 107)*x + Mod(51, 107) 1]

[Mod(1, 107)*x + Mod(80, 107) 1]

[Mod(1, 107)*x + Mod(84, 107) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 137)
%4 = 
[ Mod(1, 137)*x + Mod(7, 137) 1]

[Mod(1, 137)*x + Mod(42, 137) 1]

[Mod(1, 137)*x + Mod(89, 137) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 139)
%5 = 
[Mod(1, 139)*x + Mod(33, 139) 1]

[Mod(1, 139)*x + Mod(52, 139) 1]

[Mod(1, 139)*x + Mod(55, 139) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 151)
%6 = 
[Mod(1, 151)*x + Mod(21, 151) 1]

[Mod(1, 151)*x + Mod(39, 151) 1]

[Mod(1, 151)*x + Mod(92, 151) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 233)
%7 = 
[ Mod(1, 233)*x + Mod(49, 233) 1]

[Mod(1, 233)*x + Mod(191, 233) 1]

[Mod(1, 233)*x + Mod(227, 233) 1]
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

$$ 3 x^2 + 2 xy - 12 y^2 $$

 Represented (positive) primes up to  1000

     3     7    11    41    47    53    71    73    83   101
   127   149   157   173   181   197   211   223   229   263
   271   307   337   359   373   379   397   419   433   443
   509   521   571   593   599   613   617   619   641   659
   673   677   719   733   739   743   751   761   773   787
   811   821   887   937   953   983

parisize = 4000000, primelimit = 500000
? factormod( x^3 + x^2 - 3 * x - 1, 3)
%1 = 
[Mod(1, 3)*x^3 + Mod(1, 3)*x^2 + Mod(2, 3) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 7)
%2 = 
[Mod(1, 7)*x^3 + Mod(1, 7)*x^2 + Mod(4, 7)*x + Mod(6, 7) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 11)
%3 = 
[Mod(1, 11)*x^3 + Mod(1, 11)*x^2 + Mod(8, 11)*x + Mod(10, 11) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 41)
%4 = 
[Mod(1, 41)*x^3 + Mod(1, 41)*x^2 + Mod(38, 41)*x + Mod(40, 41) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 47)
%5 = 
[Mod(1, 47)*x^3 + Mod(1, 47)*x^2 + Mod(44, 47)*x + Mod(46, 47) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 53)
%6 = 
[Mod(1, 53)*x^3 + Mod(1, 53)*x^2 + Mod(50, 53)*x + Mod(52, 53) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 71)
%7 = 
[Mod(1, 71)*x^3 + Mod(1, 71)*x^2 + Mod(68, 71)*x + Mod(70, 71) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 73)
%8 = 
[Mod(1, 73)*x^3 + Mod(1, 73)*x^2 + Mod(70, 73)*x + Mod(72, 73) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 83)
%9 = 
[Mod(1, 83)*x^3 + Mod(1, 83)*x^2 + Mod(80, 83)*x + Mod(82, 83) 1]

? factormod( x^3 + x^2 - 3 * x - 1, 101)
%10 = 
[Mod(1, 101)*x^3 + Mod(1, 101)*x^2 + Mod(98, 101)*x + Mod(100, 101) 1]

?