Thanks for that. As it happens it's not a problem because the first 30 all have even digits and so can't be a final segment of a left-truncatable prime, except 7519441, and 441 isn't prime.
I could write a more comprehensive test - didn't seem worth it for this case, but it's an interesting exercise. It's easy to extend this test so the list of primes used goes as far as the smallest factor in each of these factorisations, say, 109. Then these would all be correctly identified as composite. It would also be easy to combine it with Pollard Rho or PPP tests, to extend the range before we need to do something more serious.
Yes the full program for left-truncatable primes happens to be fine (no false positives in its output for any length, as I verified with a different primality test).
The primality test in the post (Fermat's test) is interesting to analyze though. BTW how did you generate your list of false positives? I didn't mention it earlier, but as a shortcut I used the list from https://oeis.org/A001567 (the base-2 pseudoprimes up to 10^12), and found that the function in the post has 2690 of these false positives less than 10^12. I wrote out only the first 10 above (after piping through `factor`). Of these 2690 false positives, these have the 5 largest smallest prime factors:
And indeed the `small_primes` used by the test needs to be extended up to the prime 6917, if we want to get no false positives at least for numbers up to 10^12. And even if we went that far, we'd encounter the next false positive soon after, at 1025414056801 (factors 7561 8101 16741) and then (if we included primes up to 7561) at 1196508858409 (factors 8059 11833 12547). (See https://oeis.org/A083876)
I think compared to the Fermat test used in the post, the Miller-Rabin test is almost as easy to write (only a few lines longer), and it comes with stronger guarantees. Not only is it a test used in the real world (using it with just a few random bases gives a high degree of confidence), but it also comes with guarantees when a deterministic set of primes (like your idea) is used: for example with the first 13 primes (2 to 41), the first false positive is over 3.3 × 10^24 (https://en.wikipedia.org/w/index.php?title=Miller%E2%80%93Ra...). See also https://miller-rabin.appspot.com/
For what it's worth, here is an implementation of the Miller-Rabin primality test that first tries the smallest 13 primes and then tries some random bases:
def is_probable_prime(n, num_random_trials=5):
"""
An implementation of the Miller-Rabin primality test.
If this function returns False, then n is definitely not prime.
If this function returns True, then n is probably prime:
the probability of n being composite is less than 1/4^(num_random_trials), and
nonzero only if n >= 3317044064679887385961981 > 3.3*10^24 > 1.3*2^81 (by https://arxiv.org/abs/1509.00864).
"""
assert n == abs(int(n)), 'Need a nonnegative integer, got %s' % n
if n < 2: return False
if n % 2 == 0: return n == 2
(s, d) = (0, n - 1)
while d % 2 == 0: (s, d) = (s + 1, d // 2)
assert n - 1 == (2**s) * d and d % 2 == 1
small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
for i in range(len(small_primes) + num_random_trials):
a = small_primes[i] if i < len(small_primes) else random.randrange(small_primes[-1], n - 1)
if a >= n: break
if not (pow(a, d, n) == 1 or any(pow(a, 2**j * d, n) == n - 1 for j in range(s))):
return False
return True
Hi - thanks for the response. This will be brief, as I'm very short of time.
> Yes the full program for left-truncatable primes happens to be fine ...
Yes - although it's a little lucky, but easy enough to run an extra confirmation.
> how did you generate your list of false positives?
Nothing clever - I literally wrote a slow trial division program, then ran through all integers up to some limit - 10^8 I think - and when the "is_prime" routine said it was prime I called the slow version to see if it was right. Machines are fast.
> the function in the post has 2690 of these false positives less than 10^12. ... Of these 2690 false positives, these have the 5 largest smallest prime factors:
> And indeed the `small_primes` used by the test needs to be extended up to the prime 6917, if we want to get no false positives at least for numbers up to 10^12.
We can augment this with the Perrin Psuedo-Prime test, but Miller-Rabin is still probably the best way to go to do it "properly". If I re-wrote the material on the Perrin tests[0] I'd probably use M-R.
So we're pulling in the same direction, although making different trade-offs. And it's fun to play and see what happens.
