Find the greatest prime less than n, with n = ~10^230

Question

Is there something wrong with my solution to find the largest prime less than n when n can be up to ~10^230? Are there any suggestions for a better approach?

Here is my attempt, using the following version of a Miller-Rabin primality test in Python:

from random import randrange

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,
    547,557,563,569,571,577,587,593,599,601,
    607,613,617,619,631,641,643,647,653,659,
    661,673,677,683,691,701,709,719,727,733,
    739,743,751,757,761,769,773,787,797,809,
    811,821,823,827,829,839,853,857,859,863,
    877,881,883,887,907,911,919,929,937,941,
    947,953,967,971,977,983,991,997
]

def probably_prime(n, k):
    """Return True if n passes k rounds of the Miller-Rabin primality
    test (and is probably prime). Return False if n is proved to be
    composite.

    """
    if n < 2: return False
    for p in small_primes:
        if n < p * p: return True
        if n % p == 0: return False
    r, s = 0, n - 1
    while s % 2 == 0:
        r += 1
        s //= 2
    for _ in range(k):
        a = randrange(2, n - 1)
        x = pow(a, s, n)
        if x == 1 or x == n - 1:
            continue
        for _ in range(r - 1):
            x = pow(x, 2, n)
            if x == n - 1:
                break
            else:
                return False
    return True

I start by testing probably_prime(n) where I decrement and test each value of n until I get a "probably prime" number. When I test this on values of n = ~10^230, I am finding primes about 20-30 numbers apart. After reading more about prime gap , my results seem highly unlikely, as I should not be finding primes so frequently. I have tested k values of up to 50,000, and I am getting the same answer. What am I doing wrong, and are there any suggestions for a better solution?

Answer 1

You're right, your code seems to be having difficulty as soon as it gets beyond the small_primes table. Looking more closely, there's an error here:

    for _ in range(r - 1):
        x = pow(x, 2, n)
        if x == n - 1:
            break
        else:
            return False

You want to return False (ie composite) if you never find x == n-1 (or you can short-circuit and return False if x == 1 , I think: see here ). This can be done simply by changing the indentation:

    for _ in range(r - 1):
        x = pow(x, 2, n)
        if x == n - 1:
            break
    else:
        return False

(The for/else combination is really for/if-not-break .)

After making this change, I get:

>>> sum(orig(p, 20) for p in range(10**6, 2*10**6))
54745
>>> sum(fixed(p, 20) for p in range(10**6, 2*10**6))
70435
>>> sum(orig(p, 20) for p in range(10**230, 10**230+10**3))
40
>>> sum(fixed(p, 20) for p in range(10**230, 10**230+10**3))
2

which is correct.