Solve problems 129 and 130.

python/e129.py

@@ -0,0 +1,58 @@
+from math import gcd
+from functools import lru_cache
+
+
+@lru_cache
+def r(n):
+    assert n > 0
+    if n == 1:
+        return 1
+    else:
+        return 10**(n - 1) + r(n - 1)
+
+
+def r_closed_form(n):
+    assert n > 0
+    return (10**n - 1) // 9
+
+
+def r_modulo_closed_form(n, m):
+    assert n > 0 and m > 0
+    return ((pow(10, n, 9 * m) - 1) // 9) % m
+
+
+def a(n):
+    assert gcd(n, 10) == 1
+    k = 1
+    while True:
+        if r_modulo_closed_form(k, n) == 0:
+            return k
+        k += 1
+
+
+def euler_129():
+
+    # Comparing naiv to efficient implementations to find potential bugs.
+    for n in range(5, 1000):
+        assert r(n) == r_closed_form(n)
+        for m in range(2, 20):
+            assert (r_closed_form(n) % m)  == r_modulo_closed_form(n, m)
+
+    assert a(7) == 6
+    assert a(41) == 5
+    assert a(17) == 16
+
+    target = 10**6
+    delta = 200
+    for n in range(target, target + delta):
+        if gcd(n, 10) == 1:
+            if (av := a(n)):
+                if av > target:
+                    return n
+    return None
+
+
+if __name__ == "__main__":
+    solution = euler_129()
+    print("e129.py: " + str(solution))
+    assert(solution == 1000023)

python/e130.py

@@ -0,0 +1,32 @@
+from math import gcd
+from lib_prime import is_prime_rabin_miller
+
+
+def r_modulo_closed_form(n, m):
+    assert n > 0 and m > 0
+    return ((pow(10, n, 9 * m) - 1) // 9) % m
+
+
+def a_special(n):
+    k = n - 1
+    if r_modulo_closed_form(k, n) == 0:
+        return k
+    return None
+
+
+def euler_130():
+    c, r = 0, 0
+    for n in range(2, 10**9):
+        if gcd(n, 10) == 1 and not is_prime_rabin_miller(n):
+            if a_special(n) is not None:
+                r += n
+                c += 1
+            if c == 25:
+                break
+    return r
+
+
+if __name__ == "__main__":
+    solution = euler_130()
+    print("e130.py: " + str(solution))
+    assert(solution == 149253)