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    <p>Your observation seems to be accurate, which is quite a good catch.</p> <p>I suspect the reason that it works, at least in some cases, is because composite numbers are actually factored into multiple primes. So, the inner loop may miss the value on the first factor, but it then picks it up on a later factor.</p> <p>For a small'ish "n", you can print out values of the list to see if this is what is happening.</p> <p>This method of finding primes, by the way, is based on the Sieve of Eratothenes. It is possible when doing the sieve that if "c" is a multiple of "p", then the next value is never a multiple of the same prime.</p> <p>The question is: are there any cases where all values between p*x and p*(x+1) are divisible by some prime less than p and p*x+1). (This is where the algorithm would miss a value and it would not be caught later.) However, one of these values is even, so it would be eliminated on round "2". So, the real question is whether there are cases where all values between p*x and p*(x+2) are divisible by numbers less than p.</p> <p>Off hand, I can't think of any numbers less than 100 that meet this condition. For p = 5, there is always a value that is not divisible by 2 or 3 between two consecutive multiples of 5.</p> <p>There seems to be a lot written on prime gaps and sequences, but not so much on sequences of consecutive integers divisible by numbers less than p. After some (okay, a lot) of trial and error, I've determined that every number between 39,474 (17*2,322) and 39,491 (17*2,233) is divisible by an integer less than 17:</p> <pre><code>39,475 5 39,476 2 39,477 3 39,478 2 39,479 11 39,480 2 39,481 13 39,482 2 39,483 3 39,484 2 39,485 5 39,486 2 39,487 7 39,488 2 39,489 3 39,490 2 </code></pre> <p>I am not sure if this is the first such value. However, we would have to find sequences twice as long as this. I think that is unlikely, but not sure if there is a proof.</p> <p>My conclusion is that the original code might work, but that your fix is the right thing to do. Without a proof that there are no such sequences, it looks like a bug, albeit a bug that could be very, very, very rare.</p>
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    1. PODoes this prime function actually work?
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    1. USGordon Linoff
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    1. COYeah I'm not sure how to prove this yet. I have a deleted answer that's basically like user448810's which is wrong on further reflection.
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    2. CO"are there any cases where all values between p*x and p*(x+1) are divisible by some prime less than p?" - There shouldn't be, since all multiples of all primes less than p have been removed.
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    3. COBut 39,474 would be dropped when we are checking for % 2 == 0, this example would never show up in OPs problem.
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