Solve 293 and 686

python/e293.py

@@ -0,0 +1,70 @@
+from lib_prime import primes, is_prime_rabin_miller
+from math import isqrt
+
+
+def next_larger_prime(n):
+    """ Returns the next prime larger or equal to n. """
+    if n == 1:
+        return 2
+
+    n += 1
+    if n % 2 == 0:
+        n += 1
+
+
+    for _ in range(1000):
+        if is_prime_rabin_miller(n):
+            return n
+        n += 2
+    assert False
+
+
+def m(n):
+    higher_prime = next_larger_prime(n + 1)
+    m = higher_prime - n
+    return m
+
+
+def admissible(xs, threshold):
+    found = set()
+    numbers = [(xs[0], 0)]
+
+    while numbers:
+        value, index = numbers.pop()
+        if value >= threshold:
+            continue
+        if index == len(xs) - 1:
+            found.add(value)
+        numbers.append((value * xs[index], index))
+        if index < len(xs) - 1:
+            numbers.append((value * xs[index + 1], index + 1))
+    return found
+
+
+def euler_293():
+    n = 10**9
+    assert next_larger_prime(1) == 2
+    assert next_larger_prime(7) == 11
+    assert m(630) == 11
+
+    m_set = set()
+    ps = primes(isqrt(n) + 10*8)
+
+    pow2 = 2
+    while pow2 < n:
+        mv = m(pow2)
+        m_set.add(mv)
+        pow2 *= 2
+
+    for i in range(len(ps)):
+        xs = ps[0:i + 1]
+        for x in admissible(xs, n):
+            m_set.add(m(x))
+
+    return sum(m_set)
+
+
+if __name__ == "__main__":
+    solution = euler_293()
+    print("e293.py: " + str(solution))
+    assert solution == 2209

python/e686.py

@@ -0,0 +1,29 @@
+def p(leading_digits, count):
+    n_cutoff = 18
+    leading_digits_str = str(leading_digits)
+    n_digits = len(leading_digits_str)
+    j, current_leading, found = 1, 2, 0
+
+    while found < count:
+        current_leading *= 2
+        j += 1
+        current_leading_str = str(current_leading)
+        if  current_leading_str[:n_digits] == leading_digits_str:
+            found += 1
+        current_leading = int(current_leading_str[:n_cutoff])
+    return j
+
+
+def euler_686():
+    assert p(12, 1) == 7
+    assert p(12, 2) == 80
+    assert p(123, 45) == 12710
+    assert p(123, 100) == 28343
+    assert p(123, 1000) == 284168
+    return p(123, 678910)
+
+
+if __name__ == "__main__":
+    solution = euler_686()
+    print("e686.py: " + str(solution))
+    assert solution == 193060223