class number and negative Pell equation

Question

Consider odd $d$ with no prime factor $4n+3$. For many but not all such $d$, the negative Pell equation is solvable, i.e. the period of (the continued fraction of) $d$ is odd. (See oeis A249052 for examples, starting with 205.) If the class number $h$ (of ${\mathbb Q}(\sqrt d)$) is 1, then $h = 1$ and odd period implies $d$ is prime (as one can show using textbook methods), and (without assumption on $h$), if $d$ is a prime-power then negative Pell is solvable (Perron p. 108). Now suppose $h = 1$ and $d$ has no prime factor $4n+3$. Then does it follow that the period is odd and $d$ is prime? (As remarked above, those last two conditions are equivalent when $h=1$.) That is, all the numbers in A249052 have class number greater than 1?
It is numerically true for $d < 100,000$. That "does it follow" is the first part of my question.

We know there must be some intimate connection between the solvability of the negative Pell equation and the class number, since it is solvable if and only if the strict class number equals the class number. So the second part of my question is whether there exists some formula like the class number formula but involving the strict class number instead of the class number.

Answer 1

Added: my indefinite forms are reduced in the sense of Gauss and Lagrange. That is, $\langle a, b, c \rangle$ means the form $f(x,y) = a x^2 + b xy + c y^2$ with discriminant $\Delta = b^2 - 4ac.$ Primitive means $\gcd(a,b,c)=1$ while reduced means $$ ac < 0 \; , \; \; and \; \; \; \; b > |a+c|. $$ There is a proof that this is equivalent to the original "reduced" criteria in a book of Franz Lemmermeyer. I found it by repreated fiddling with the Conway topograph and the Gauss-Lagrange method of neighboring forms.

Here are some examples. Here the ideal class number is halfmy form class number. Simply put, my $\langle 1, 13, -9 \rangle$ and $\langle -1, 13, 9 \rangle$ are mapped to the same ideal. This is all in Buell and other books. Now, consider the (positive) primes represented by $\langle 3, 13, -3 \rangle.$ Evident is $3,$ which is a nonresidue mod 5 and mod 41 because $5,41 \equiv 3 \pmod 3$

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205 factored 5 * 41

1.             1          13          -9   cycle length             4
2.            -1          13           9   cycle length             4
3.             3          13          -3   cycle length             4
4.            -3          13           3   cycle length             4

form class number is 4

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jagy@phobeusjunior:~$ ./Conway_Positive_Primes 3 13 -3 500
3 13 -3 original form 3 13 -3 Lagrange-Gauss reduced

Represented (positive) primes up to 500

     3     7    13    17    47    53    67    97   137   157
   167   193   227   233   257   263   293   313   317   347
   383   397   457   463

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these are the collection of remainders when dividing by 5

  2      3

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these are the collection of remainders when dividing by 41

  3      6      7     11     12     13     14     15     17     19
 22     26     28     29     30     34

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221 factored 13 * 17

1.             1          13         -13   cycle length             2
2.            -1          13          13   cycle length             2
3.             5          11          -5   cycle length             4
4.            -5          11           5   cycle length             4

form class number is 4

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5945 factored 5 * 29 * 41

1.             1          77          -4   cycle length            10
2.            -1          77           4   cycle length            10
3.             2          77          -2   cycle length            14
4.            -2          77           2   cycle length            14
5.             7          73         -22   cycle length            20
6.            -7          73          22   cycle length            20
7.            11          73         -14   cycle length            20
8.           -11          73          14   cycle length            20

form class number is 8

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jagy@phobeusjunior:~/Desktop/Cplusplus$ ./indefCycleLeft  1 77 -4

0  form   1 77 -4   epsilon  77     ambiguous  
1  form   -4 77 1   Epsilon  -19     ambiguous   opposite  
2  form   20 75 -4   Epsilon  3
3  form   -49 45 20   Epsilon  -1
4  form   16 53 -49   Epsilon  4
5  form   -5 75 16   Epsilon  -15     ambiguous  
6  form   16 75 -5   Epsilon  4     ambiguous  
7  form   -49 53 16   Epsilon  -1
8  form   20 45 -49   Epsilon  3
9  form   -4 75 20   Epsilon  -19
10  form   1 77 -4


  form   1 x^2  + 77 x y  -4 y^2 

minimum was   1rep   x = 1   y = 0 disc 5945 dSqrt 77
Automorph, written on right of Gram matrix:  
-159166445  8262816
2065704  -107237
  for   Pari/gp: rt =  [ -159166445 , 2065704 ; 8262816 , -107237 ] ;    h =  [ 2 , 77 ; 77 , -8 ] ;    r =  [ -159166445 , 8262816 ; 2065704 , -107237 ] ; 


 opposite Pari/gp: rt =  [ 77 , -1 ; 1 , 0 ] ;    h =  [ 2 , 77 ; 77 , -8 ] ;    r =  [ 77 , 1 ; -1 , 0 ] ; 

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