Created
March 12, 2026 20:42
-
-
Save Hermann-SW/b538c991c922b1e8fc06abbab554b3ef to your computer and use it in GitHub Desktop.
SageMath diophantine S-Unit solve example: error free after many iterations of Google Gemini; changed to ℚ and added generator output by me
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| # 1. Setup | |
| x = polygen(QQ, 'x') | |
| # K.<i> = NumberField(x^2 + 1) # ℚ(i) | |
| K.<i> = NumberField(x - 1) | |
| # Using this, the root a is just 1. This forces Sage to wrap the rational | |
| # numbers in a "NumberField object" which possesses the .S_unit_group() method. | |
| # | |
| S_list = K.primes_above(2) + K.primes_above(3) | |
| S = tuple(S_list) | |
| # 2. Get the actual generators of the S-unit group | |
| U_S = K.S_unit_group(S=S) | |
| gens = U_S.gens() | |
| print("Generators:") | |
| for g in U_S.gens_values(): | |
| print(g) | |
| # 3. Solve | |
| from sage.rings.number_field.S_unit_solver import solve_S_unit_equation | |
| solutions = solve_S_unit_equation(K, S) | |
| def vector_to_number(vec, generators): | |
| """Manually multiply generators by their exponents.""" | |
| res = K(1) | |
| for i in range(len(vec)): | |
| # We convert the generator to the field K to be safe | |
| res *= K(generators[i])**vec[i] | |
| return res | |
| print(f"Found {len(solutions)} solutions:") | |
| for entry in solutions: | |
| # entry[0] and entry[1] are the exponent vectors | |
| x_val = vector_to_number(entry[0], gens) | |
| y_val = vector_to_number(entry[1], gens) | |
| print(f"{x_val} + {y_val} == {x_val + y_val}") | |
| print("-" * 20) |
Author
Author
Now tried with S={2,3,5}, hoping for better runtime than with {2,3,5,7} given that S={2,3} takes less than 5 minutes.
I aborted after 13.5h of 5.4GHz boost frequency run on AMD 7950X CPU:
hermann@7950x:~$ time Downloads/SageMath-10.8-x86_64.AppImage S-Unit.sol3.sage
Generators:
-1
2
3
5
^CTraceback (most recent call last):
File "/tmp/.mount_SageMaKhnGdE/app_root/local/bin/sage", line 5, in <module>
from sage.cli.__main__ import main
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/cli/__main__.py", line 9, in <module>
sys.exit(main())
~~~~^^
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/cli/__init__.py", line 56, in main
return RunFileCmd(options).run()
~~~~~~~~~~~~~~~~~~~~~~~^^
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/cli/run_file_cmd.py", line 50, in run
eval(compile(open(input_file, 'rb').read(), input_file, 'exec'), sage_globals())
~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/tmp/sage_zm6ra6cv/S-Unit.sol3bjhh8xix.sage.py", line 27, in <module>
solutions = solve_S_unit_equation(K, S)
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/rings/number_field/S_unit_solver.py", line 2835, in solve_S_unit_equation
S_unit_solutions = sieve_below_bound(K, list(S), final_LLL_bound, verbose=verbose)
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/rings/number_field/S_unit_solver.py", line 2718, in sieve_below_bound
cs_list = compatible_systems(split_primes_list, complement_exp_vec_dict)
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/rings/number_field/S_unit_solver.py", line 2489, in compatible_systems
old_systems = compatible_systems(S1, complement_exp_vec_dict)
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/rings/number_field/S_unit_solver.py", line 2489, in compatible_systems
old_systems = compatible_systems(S1, complement_exp_vec_dict)
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/rings/number_field/S_unit_solver.py", line 2496, in compatible_systems
if all((compatible_vectors_check(exp_vec, exp_vec_qj, g, l) and
~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
compatible_vectors_check(comp_vec, comp_vec_qj, g, l))
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
for g, (exp_vec_qj, comp_vec_qj) in zip(gcds, old_system)):
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/var/tmp/sage-10.8/local/lib/python3.13/site-packages/sage/rings/number_field/S_unit_solver.py", line 2496, in <genexpr>
if all((compatible_vectors_check(exp_vec, exp_vec_qj, g, l) and
File "cysignals/signals.pyx", line 357, in cysignals.signals.python_check_interrupt
KeyboardInterrupt
^C
real 831m40.662s
user 831m35.096s
sys 0m2.705s
hermann@7950x:~$
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Instead S={2,3} using S={2,3,5,7} ...
... leads to runtime explosion: aborted after 22.5h single core on AMD 7950X CPU with boost frequency of 5.4GHz: