Solve problem 800

python/e800.py

@@ -0,0 +1,68 @@
+from lib_prime import primes
+from math import log, ceil
+
+
+def c(n_base: int) -> int:
+    """Compute the number of prime pairs (p, q) where
+    p**q * q**p <= n_base**n_base and p < q"""
+    digits = ceil(log(n_base, 10)) + 1
+    n = n_base**n_base
+    n_log = log(n)
+    ps = primes(2 * 10**digits)
+    p, q = ps[0], ps[-1]
+    assert (p**q * q**p) > n, "largest prime too low"
+    j_max = len(ps) - 1
+
+    t = 0
+    for i in range(len(ps)):
+        p, q = ps[i], ps[i + 1]
+
+        # Crux:
+        #             (p**q * q**p) > n
+        # <=>       ln(p**q * q**p) > ln(n)
+        # <=>   ln(p**q) + ln(q**p) > ln(n)
+        # <=> q * ln(p) + p * ln(q) > ln(n)
+        # Much lower numbers and therefore faster to compute.
+        if q * log(p) + p * log(q) > n_log:
+            break
+
+        # Binary search to find the largest j for which p**q * q**p <= n:
+        j_lo, j_hi = i + 1, j_max
+        while (j_hi - j_lo) > 1:
+            j_mid = (j_hi + j_lo) // 2
+            p, q = ps[i], ps[j_mid]
+            if q * log(p) + p * log(q) > n_log:
+                j_hi = j_mid
+            else:
+                j_lo = j_mid
+        j_max = j_lo + 1
+
+        # All primes lower than ps[j_lo] have to be counted.
+        t += j_lo - i
+
+        # p_ref, q_ref = ps[i], ps[j_lo]
+        # print(p_ref, q_ref, "<=")
+
+        # Turns out the binary search is not necessary with the optimized
+        # version of the comparison and the following works just as fine.
+        # for j in range(j_max, i, -1):
+        #     p, q = ps[i], ps[j]
+        #     if q * log(p) + p * log(q) > n_log:
+        #         j_max = j
+        #     else:
+        #         assert p == p_ref
+        #         assert q == q_ref
+        #         t += (j - i)
+        #         break
+    return t
+
+
+def euler_800():
+    assert c(800) == 10790
+    return c(800800)
+
+
+if __name__ == "__main__":
+    solution = euler_800()
+    print(f"e800.py: {solution}")
+    assert(solution == 1412403576)