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Euler famously showed that there are at least 65 idoneal (convenient) numbers. This was Euler's definition of idoneal number:

The number $n$ is idoneal if the following holds: Let $m>1$ be an odd number relatively prime to n which can be written in the form $x^2+ny^2$ with $x,y$ relatively prime. If the equation $m=x^2+ny^2$ has only one solution with $x,y≥0$, then $m$ is a prime number.

Moreover, it is allegedly said it exists infinitely many primes for each idoneal number. Is it true ?

With $n=k^2$ an even square, does that mean that $x,y$ are odd numbers ? So, with $k=1$, it exists infinitely many primes m with $m=x^2+4y^2$ and with $x,y$ odd numbers ?

francois
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  • @lulu If I am informed right , GRH implies that the list is complete and it is unconditionally proven that at most two more idoneal numbers exist. But that does not prevent us to establish (at least in principle) a proof using idoneal numbers for infinite many primes. I think , in fact , infinite many primes could be proven to be prime this way. – Peter Sep 03 '23 at 14:29
  • @Peter Maybe I don't understand. How could a single (or a finite list) of such numbers give us infinitely many primes? Is there an example using $1$, say? – lulu Sep 03 '23 at 14:49
  • @lulu Infinite many primes that can be proven to be prime by using the representations of $x^2+dy^2$ , where $d$ is idoneal. This is however an extremely impractical approach and morevoer does not work for every prime. – Peter Sep 03 '23 at 14:51
  • Example $d=1$ : Every prime $p$ of the form $4k+1$ has a unique representation $x^2+y^2=p$ and the fact that this representation is unique can be used to prove that $p$ is prime. – Peter Sep 03 '23 at 14:54
  • @Peter Ok, so one might show that there are infinitely many primes representable as $x^2+y^2$ uniquely (i.e., those $\equiv 1 \pmod 4$). I guess you could do that by looking at the norm from $\mathbb Z[i]$. Fair enough. I'll delete my first comment accordingly. – lulu Sep 03 '23 at 14:54

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The form $4 x^2 + 4xy + 5 y^2$ represents infinitely many primes (in the long run, one quarter of them). See the book by Cox, Primes of the Form $x^2 + n y^2.$

Next, to get primes we need that $y$ to be odd. Finally $$ 4 x^2 + 4 xy + 5 y^2 = (2x+y)^2 + 4 y^2. $$

Will Jagy
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  • Thank you @Jagy. I have the book by Cox. Which is the page number, please ? – francois Sep 03 '23 at 16:47
  • Page 28-29, we have $4x^2+4xy+(2^r+1)y^2$. And it works with r=2. – francois Sep 03 '23 at 18:42
  • @francois The other description of idoneal is: one form per genus. Later in the book, he shows how the forms in a genus each represents a certain fraction of the primes that are permitted for the genus. So, for examples, together $x^2 + 14 y^2$and $2 x^2 + 7 y^2$ represent all primes permitted $\pmod 8$ and $\pmod 7.$ However, it is harder to describe which primes go with which form. In the first edition, this is Theorem 5.33 on page 115. Main result is Theorem 9.12, page 188 – Will Jagy Sep 03 '23 at 19:17