Problem 3: Prime time

def is_prime(n): """Return True iff n is prime. Works for arbitrarily-large ints (uses deterministic Miller-Rabin with bases that are proven complete for every 64-bit input, and then additional bases for bigger numbers).""" # Handle tiny inputs quickly. if n < 2: return False small = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37) for p in small: if n % p == 0: return n == p # Decompose n-1 as d * 2^s d, s = n - 1, 0 while d % 2 == 0: d //= 2 s += 1 def _try(a): """Miller-Rabin witness loop for base a.""" x = pow(a, d, n) if x == 1 or x == n - 1: return False for _ in range(s - 1): x = pow(x, 2, n) if x == n - 1: return False return True # composite # Bases that make the test deterministic for every 64-bit number bases64 = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37) if n < 341550071728321: # < 3.4e14 bases = (2, 3, 5, 7, 11, 13, 17) elif n < 3825123056546413051: # < 3.8e18 bases = (2, 3, 5, 7, 11, 13, 17, 19, 23) else: # full 64-bit range bases = bases64 for a in bases: if _try(a): return False return True # ------------------------------------------------------------------ # Quick self-test (executed only when this file is run directly). # ------------------------------------------------------------------ if __name__ == "__main__": small_primes = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541 } # Exhaustive test up to 541 for x in range(max(small_primes) + 1): assert is_prime(x) == (x in small_primes) # Two big-integer assertions from the statement assert is_prime(50944352050432840594664327703654217607483249127368282443658421370320240418632150463386971396202568194925235923024411275631368759242814890471657334733218208416824863730606520908198845521423439226153813263186188880157525787115965153854071772222183528102751911) assert not is_prime(270232216891381492844952749373617571683872847618266214418525494195836938837070446176769548415673569982361014481475276843062714955758548211652446400421420170897612445799329432093790003911616167788064934840566031517914337645532465618728202484917009287562085840204296939771978314429077070675423479) print("All tests passed.")