feat(blockchain): add all_diophantine_solutions with type hints & doctests (#12999)

* feat(blockchain): add all_diophantine_solutions with type hints & doctests

* feat(blockchain): add all_diophantine_solutions; move above __main__, remove future import

Author: Zhanbingli

blockchain/diophantine_equation.py

@@ -100,6 +100,64 @@ def extended_gcd(a: int, b: int) -> tuple[int, int, int]:
     return (d, x, y)
 
 
+def all_diophantine_solutions(
+    a: int,
+    b: int,
+    c: int,
+    n: int = 2,
+) -> list[tuple[int, int]]:
+    """
+    Return up to `n` integer solutions (x, y) to the linear Diophantine equation
+    a*x + b*y = c using the extended Euclidean algorithm.
+
+    Raises
+    ------
+    ValueError
+        If no integer solutions exist.
+
+    Time complexity
+    ---------------
+    O(log(max(|a|, |b|))) to compute a base solution using extended_gcd;
+    plus O(n) to enumerate `n` solutions.
+
+    Space complexity
+    ----------------
+    O(1) beyond the returned list.
+
+    Examples
+    --------
+    >>> all_diophantine_solutions(10, 6, 14, n=2)
+    [(-7, 14), (-4, 9)]
+    >>> all_diophantine_solutions(10, 6, 14, n=4)
+    [(-7, 14), (-4, 9), (-1, 4), (2, -1)]
+    >>> all_diophantine_solutions(3, 6, 10, n=1)
+    Traceback (most recent call last):
+    ...
+    ValueError: No integer solutions exist for a=3, b=6, c=10
+    """
+    if a == 0 and b == 0:
+        if c == 0:
+            # Infinite solutions; return one canonical solution.
+            return [(0, 0)][: min(1, n)]
+        raise ValueError("No integer solutions exist for a=0, b=0, c!=0")
+
+    g, xg, yg = extended_gcd(abs(a), abs(b))
+    if c % g != 0:
+        msg = f"No integer solutions exist for a={a}, b={b}, c={c}"
+        raise ValueError(msg)
+
+    # Scale a particular solution to ax + by = c
+    x0, y0 = xg * (c // g), yg * (c // g)
+    if a < 0:
+        x0 = -x0
+    if b < 0:
+        y0 = -y0
+
+    # General solution: x = x0 + t*(b/g), y = y0 - t*(a/g)
+    dx, dy = b // g, a // g
+    return [(x0 + t * dx, y0 - t * dy) for t in range(n)]
+
+
 if __name__ == "__main__":
     from doctest import testmod