Proving that the Diophantine equation $(11a + 5b)^2 - 223b^2 = \pm 11$ has no solutions

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.

• $11$ is not a quadratic residue modulo $223$ – J. W. Tanner Aug 5 '20 at 1:25
• $3$ is not either, but $-11$ and $-3$ are. Anyway, these look like generalized Pell equations – J. W. Tanner Aug 5 '20 at 1:45
• 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. – Andrew Tindall Aug 5 '20 at 1:51

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 $$33$$.

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
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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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