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I am working on an algebraic number theory exercise, which is to prove that $\mathbb Z[\sqrt{223}]$ has three ideal classes. I've run against the following two (really four) Diophantine equations: $$ (11a + 5b)^2 - 223b^2 = \pm 11 $$

$$ (3a + b)^2 - 223b^2 = \pm 3 $$

I think I should be able to prove that neither of these pairs of equations has any solutions in $\mathbb Z^2$ - I have run a program to check all small values of $a$ and $b$ (less than 10,000) and found nothing, but I know that the minimum solutions to equations like this can be quite large.

What I've tried doing so far is reducing the first equation mod $11$ and mod $5$, both of which seem to give tautologies, and reducing the second equation mod $3$, which also wasn't useful. I don't know much in this area, so I'm not sure how to begin attacking the problem.

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    $\begingroup$ $11$ is not a quadratic residue modulo $223$ $\endgroup$ – J. W. Tanner Aug 5 '20 at 1:25
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    $\begingroup$ $3$ is not either, but $-11$ and $-3$ are. Anyway, these look like generalized Pell equations $\endgroup$ – J. W. Tanner Aug 5 '20 at 1:45
  • $\begingroup$ No, but -11 is: 99^2 = -11 mod 223. It looks like Conrad's notes give an effective bound for searching, at least once you've got a solution to the corresponding integral equation. I'll try that direction. $\endgroup$ – Andrew Tindall Aug 5 '20 at 1:51
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There are techniques due to Dirichlet that achieve what you want in a finite number of steps. In the present case, the following ad-hoc computations do the trick.

First observe that $\alpha = 14 + \sqrt{223}$ has norm $-27$ (this implies that your second equation has a rational solution, which in turn suggests that you cannot prove it impossible by working modulo integers). Thus if there is an element of nurm $\pm 3$, one of the elements $\alpha$, $\varepsilon \alpha$ or $\varepsilon^2\alpha$ must be a cube, where $\varepsilon = 224 + 15 \sqrt{223}$ is the fundamental unit (which can be computed from the element $\beta = 15 + \sqrt{223}$ with norm $2$ via $\varepsilon = \beta^2/2$). Now you check that none of these elements is a cube.

For showing that $\alpha$ is not a cube assume that $\alpha = \gamma^3$ and $\alpha' = {\gamma'}^3$. Then $\gamma \approx 3.07$ and $\gamma' \approx -0,977$, and since $\gamma + \gamma'$ is not an integer, this is impossible.

The ideals of norm $11$ do not contribute to the class group since $16 \pm \sqrt{223}$ have norm $33$.

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The mapping from binary quadratic forms to ideals is dealt with in Henri Cohen, A Course in Computational Algebraic Number Theory, especially section 5.2 on pages 225-230. Look at that, he does real quadratic fields in section 5.6, pages 262-269.

When the principal form does not also represent $-1,$ the mapping from form (classes) to ideals is two to one. The form class number is six, your number is three. You are making this harder than necessary. My forms are "reduced" in the sense of Gauss and Lagrange, $\langle a,b,c \rangle$ with discriminant $b^2 - 4 a c.$ Reduced is equivalent to $ac < 0 $ and $b > | a+c|.$ Good luck that all the $b$ came out the same, it makes Dirichlet's description of composition come out perfectly. I am posting the positive primes represented... However, the way I found the six classes amounts to finding the Gauss-Lagrange cycle of each form. Apparently there are 32 reduced forms of this discriminant. Two reduced forms are $SL_2 \mathbb Z$ equivalent if and only if they occur in the same cycle. here are the six cycles that account for every reduced form of this discriminant. Oh, a number $r$ with $|r| < \sqrt {223} \approx 14.93$ is primitively represented by a form if and only if it is the first or third element in one of the triples in the cycle for the form. Theorem of Lagrange.

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 28 -27

0  form   1 28 -27   delta  -1     ambiguous  
1  form   -27 26 2   delta  13
2  form   2 26 -27   delta  -1     ambiguous  
3  form   -27 28 1   delta  28
4  form   1 28 -27
  form   1 x^2  + 28 x y  -27 y^2
===========================================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle -1 28 27

0  form   -1 28 27   delta  1     ambiguous  
1  form   27 26 -2   delta  -13
2  form   -2 26 27   delta  1     ambiguous  
3  form   27 28 -1   delta  -28
4  form   -1 28 27
  form   -1 x^2  + 28 x y  27 y^2 
=======================================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 3 28 -9

0  form   3 28 -9   delta  -3
1  form   -9 26 6   delta  4
2  form   6 22 -17   delta  -1
3  form   -17 12 11   delta  1
4  form   11 10 -18   delta  -1
5  form   -18 26 3   delta  9
6  form   3 28 -9
  form   3 x^2  + 28 x y  -9 y^2 
=====================================================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle -3 28 9

0  form   -3 28 9   delta  3
1  form   9 26 -6   delta  -4
2  form   -6 22 17   delta  1
3  form   17 12 -11   delta  -1
4  form   -11 10 18   delta  1
5  form   18 26 -3   delta  -9
6  form   -3 28 9
  form   -3 x^2  + 28 x y  9 y^2 
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 9 28 -3

0  form   9 28 -3   delta  -9
1  form   -3 26 18   delta  1
2  form   18 10 -11   delta  -1
3  form   -11 12 17   delta  1
4  form   17 22 -6   delta  -4
5  form   -6 26 9   delta  3
6  form   9 28 -3
  form   9 x^2  + 28 x y  -3 y^2 
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle -9 28 3

0  form   -9 28 3   delta  9
1  form   3 26 -18   delta  -1
2  form   -18 10 11   delta  1
3  form   11 12 -17   delta  -1
4  form   -17 22 6   delta  4
5  form   6 26 -9   delta  -3
6  form   -9 28 3
  form   -9 x^2  + 28 x y  3 y^2 
=========================================

Conway's Topograph method is perfectly suited for giving an indefinite form and asking for just the positive primes represented by it. Then ask the same question for $\langle -c,b,-a \rangle$

    1.             1          28         -27   cycle length             4
    2.            -1          28          27   cycle length             4
    3.             3          28          -9   cycle length             6
    4.            -3          28           9   cycle length             6
    5.             9          28          -3   cycle length             6
    6.            -9          28           3   cycle length             6
jagy@phobeusjunior:~$ ./Conway_Positive_Primes 1 28 -27  5000   223
           1          28         -27   Lagrange-Gauss reduced 
 Represented (positive) primes up to  5000

     2   101   109   197   353   401   433   509   677   857
   997  1109  1129  1193  1381  1481  1709  1873  2069  2081
  2089  2113  2269  2357  2441  2609  2617  2693  2857  2957
  3137  3169  3253  3373  3469  3673  3701  3769  3853  3929
  4001  4057  4133  4253  4721  4733  4789  4937
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=   
jagy@phobeusjunior:~$ ./Conway_Positive_Primes 27 28 -1  5000   223
          27          28          -1   Lagrange-Gauss reduced 
 Represented (positive) primes up to  5000

    71    79   107   163   223   523   563   691   739   811
   823   859   883   919   967   983   991  1163  1223  1487
  1523  1543  1607  1787  1811  1907  1951  2003  2027  2099
  2243  2423  2647  2659  2687  2699  3083  3271  3307  3343
  3539  3559  3727  3803  3931  4139  4327  4451  4483  4519
  4547  4703  4919  4999
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=   
jagy@phobeusjunior:~$ ./Conway_Positive_Primes 3 28 -9  5000   223
           3          28          -9   Lagrange-Gauss reduced 
 Represented (positive) primes up to  5000

     3    11    23    59    67   103   151   167   191   263
   271   307   311   331   383   431   439   467   491   503
   571   587   607   619   631   787   827   839   863   971
  1039  1051  1087  1283  1291  1307  1319  1399  1423  1451
  1483  1499  1511  1531  1559  1567  1571  1583  1663  1747
  1759  1783  1871  1879  1931  1979  1999  2087  2111  2251
  2287  2347  2371  2459  2543  2711  2767  2843  2939  3067
  3079  3167  3251  3259  3331  3371  3391  3463  3467  3499
  3527  3571  3643  3659  3671  3691  3719  3967  4007  4019
  4027  4091  4099  4111  4127  4159  4219  4243  4259  4283
  4339  4391  4423  4463  4567  4583  4651  4679  4723  4787
  4951  4967
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 

jagy@phobeusjunior:~$ ./Conway_Positive_Primes 9 28 -3  5000   223
           9          28          -3   Lagrange-Gauss reduced 
 Represented (positive) primes up to  5000

    17    29    37    41    53    73    89   181   241   257
   281   317   349   389   461   577   617   673   701   733
   769   797   821   881   929   941  1013  1061  1069  1093
  1117  1153  1181  1201  1213  1277  1453  1549  1597  1621
  1637  1693  1697  1733  1801  1889  1997  2137  2153  2237
  2273  2293  2521  2677  2713  2729  2741  2749  2777  2797
  2917  3037  3061  3109  3257  3301  3361  3413  3457  3461
  3517  3533  3541  3557  3593  3617  3637  3677  3793  3821
  3877  3889  3917  4021  4129  4153  4157  4217  4241  4273
  4297  4337  4349  4357  4373  4409  4457  4493  4513  4549
  4561  4637  4657  4673  4793  4813  4861  4969
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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