# About list of discriminants of real quadratic fields with narrow class number 1?

I have a couple of questions regarding the list of discriminants of real quadratic fields with narrow class number 1.

The sequence A003655 in OEIS portraits a list of discriminants of real quadratic fields with narrow class number 1. In the sequence there is no indication that the list is complete. Q1: Is that the case? Q2: In any case, could you point to some relevant references about this list?

One further question following the first responses.

1. There are finitely many determinants $\Delta=4m>0$ with $m$ square-free and narrow class number 1. $\Delta=8$ is one such determinant, is it the only one? (references?).

Thanks in advance, and regards, Guillermo

• @Will Thanks for your post. Unfortunately, I have no immediate access to the book by Buell; hopefully my library will have it in a few days. The discussion by Cohen and Lenstra seems to be about class number 1 not narrow class number 1. Oct 11, 2011 at 6:34
• @Will and @GH Thanks a lot for your comments... Oct 12, 2011 at 3:38
• There are better places for asking basic questions like these. Oct 12, 2011 at 14:36
• @Guillermo, to your "one further question": If the narrow class number of $\Delta=4m$ is one, then $m$ is prime, and we conjecture that there are infinitely many such primes. The list begins as follows: $m=2,5,13,17,29,41$. You can read more about these things and find tables as well in Rose: A course in number theory. As Franz Lemmermeyer mentioned, your questions are not of research level. Oct 14, 2011 at 13:05
• If you need a reference not going back as far as Gauss's Disquisitiones, you can do worse than pick up Flath's Number Theory. Oct 14, 2011 at 14:20

One of the references at your oeis sequence is D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241. On page 103, he proves that the narrow class group you ask about is isomorphic to the class group of binary quadratic forms. On page 82, Buell points out that computations show about 80 percent of positive prime discriminants have class number one. Note these primes are $1 \pmod 4.$ And it is certainly conjectured that the list is infinite.

INSERTED::: I found the quote I really wanted, Buell top of page 79:

Aside from the exceptional discriminants of very small magnitude, an odd class number can only occur for an odd prime discriminant.

The Cohen-Lenstra heuristics are discussed in http://en.wikipedia.org/wiki/Class_number_problem including this quote

For real fields they predict that about 75.446% will have class number 1, a result that agrees with computations

P. S. Buell let some errors into the tables. He sent me corrections for Table 2A, odd positive fundamental discriminants.

EDIT: found it! Buell page 101, Theorem 6.19, part (b): If the discriminant $\Delta$ of the quadratic field $\mathbf Q(\sqrt \Delta)$ is positive, and a solution exists to the equation $x^2 - \Delta y^2 = -4,$ then the class group and the narrow class group are isomorphic.

Then , on page 162, Theorem 9.3, If $\Delta = p,$ with $p$ a prime congruent to 1 modulo 4, then equation (9.1) is solvable, and the class number is odd.

Meanwhile the mentioned equation is (9.1) $X^2 - \Delta Y^2 = -4$

So for positive prime discriminants $p \equiv 1 \pmod 4,$ everything agrees, and we are discussing the narrow class group in this case.

There is a simple proof that (9.1) is solvable in the case mentioned, it should be in Mordell's book on Diophantine Equations, the proof should actually be for $-1$ rather than $-4.$

EDIT TOOO: I thought I might put in the example I mentioned, discriminant 9973 (prime), I have programs that take a given indefinite binary form to "reduced" status and then show the entire cycle of reduced forms, so here is what happens for $x^2 + x y - 2493 y^2,$

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 1 1 -2493 0 form 1 99 -43 delta -2 1 form -43 73 27 delta 3 2 form 27 89 -19 delta -4 3 form -19 63 79 delta 1 4 form 79 95 -3 delta -32 5 form -3 97 47 delta 2 6 form 47 91 -9 delta -10 7 form -9 89 57 delta 1 8 form 57 25 -41 delta -1 9 form -41 57 41 delta 1 10 form 41 25 -57 delta -1 11 form -57 89 9 delta 10 12 form 9 91 -47 delta -2 13 form -47 97 3 delta 32 14 form 3 95 -79 delta -1 15 form -79 63 19 delta 4 16 form 19 89 -27 delta -3 17 form -27 73 43 delta 2 18 form 43 99 -1 delta -99 19 form -1 99 43 delta 2 20 form 43 73 -27 delta -3 21 form -27 89 19 delta 4 22 form 19 63 -79 delta -1 23 form -79 95 3 delta 32 24 form 3 97 -47 delta -2 25 form -47 91 9 delta 10 26 form 9 89 -57 delta -1 27 form -57 25 41 delta 1 28 form 41 57 -41 delta -1 29 form -41 25 57 delta 1 30 form 57 89 -9 delta -10 31 form -9 91 47 delta 2 32 form 47 97 -3 delta -32 33 form -3 95 79 delta 1 34 form 79 63 -19 delta -4 35 form -19 89 27 delta 3 36 form 27 73 -43 delta -2 37 form -43 99 1 delta 99 38 form 1 99 -43 minimum was 1rep 1 0 disc 9973 dSqrt 99.864908752 M_Ratio 9973 Automorph, written on right of Gram matrix: 1938468307 1282756488 -569466216 1395887771 Trace: -960611218 gcd(a21, a22 - a11, a12) : 8 =========================================  where one can see that halfway through the cycle (step numbered 18) we have represented$-1$as a first or third coefficient of an equivalent form. This is Buell's description of "reduced" and adjacent forms. The middle coefficient is always positive and tends to be comparatively large. THere is an extreme behavior when an odd prime is a square plus 4, as$293 = 17^2 + 4,$========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
1  1  -73

0  form   1 17 -1   delta  -17
1  form   -1 17 1   delta  17
2  form   1 17 -1
minimum was   1rep 1 0 disc   293 dSqrt 17.117242769  M_Ratio  293
Automorph, written on right of Gram matrix:
-1  -17
-17  -290
Trace:  -291   gcd(a21, a22 - a11, a12) : 17
=========================================


There may be only a finite number of odd primes $p$ with $p = u^2 + 4,$ unknown.

As GH mentioned discriminant $p = 4 u^2 + 1,$ here is one with class number one:

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
1 1  -169

0  form   1 25 -13   delta  -1
1  form   -13 1 13   delta  1
2  form   13 25 -1   delta  -25
3  form   -1 25 13   delta  1
4  form   13 1 -13   delta  -1
5  form   -13 25 1   delta  25
6  form   1 25 -13
minimum was   1rep 1 0 disc   677 dSqrt 26.019223663  M_Ratio  677
Automorph, written on right of Gram matrix:
-53  -1352
-104  -2653
Trace:  -2706   gcd(a21, a22 - a11, a12) : 104
=========================================


Finally, the first odd number where we are a little surprised at the impossibility of the negative Pell equation is 205, in that $205 = 5 \cdot 41$ is not divisible by any prime $q \equiv 3 \pmod 4$ but $x^2 - 205 y^2 = -1$ and $x^2 - 205 y^2 = -4$ are impossible,

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 1 1 -51 0 form 1 13 -9 delta -1 1 form -9 5 5 delta 1 2 form 5 5 -9 delta -1 3 form -9 13 1 delta 13 4 form 1 13 -9 minimum was 1rep 1 0 disc 205 dSqrt 14.317821063 M_Ratio 205 Automorph, written on right of Gram matrix: 2 27 3 41 Trace: 43 gcd(a21, a22 - a11, a12) : 3 =========================================  Here we are using a theorem of Lagrange, that given an indefinite form with positive nonsquare discriminant$\Delta,$then any nonzero integer$n$primitively represented by the form, also satisfying$ |n| < \frac{1}{2} \sqrt \Delta,$occurs as the first coefficient of at least one form in the cycle of adjacent equivalent reduced forms. This is Theorem 85 on page 111 of Introduction to the Theory of Numbers by Leonard Eugene Dickson (1929). I do not see that Buell proves (or even quotes) the full Lagrange result, but his Theorem 3.18 on page 42 does the result for target$-4$that we use here. An unusual, and quite pretty, description of the numbers represented by an indefinite binary is in pages 18-23 of The Sensual Quadratic Form by John Horton Conway. The presentation includes a method that quickly finds all represented numbers up to some bound in absolute value, this is the Climbing Lemma on page 11. Oh,$x^2 + x y - k y^2$represents a superset of the numbers represented by$x^2 - (4k+1) y^2.$The class number 1 problem was solved only for rather special positive fundamental discriminants. These are cases where the regulator of the underlying real quadratic field is automatically small, so that by Siegel's theorem the class number is automatically large (although non-effectively so). In particular, Biró compiled a full list of$d$'s with$h(d)=1$when$d=n^2+4$or$d=4n^2+1\$ (see Acta Arith. 106 (2003), 85–104; Acta Arith. 107 (2003), 179–194). This result strikes as a real analogue of Gauss' original class number 1 problem (for negative fundamental discriminants).