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COMMENTS
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Start with a finite list of primes of the form 6k + 5, in this case, the 1-element set {5}. If the list has an even number of primes, duplicate one of them, preferably the smallest one. Then multiply the primes on the list (sometimes the larger primes will be multiplied by 25 rather than 5) and add 6.
Thus we get another number that is either a prime of the form 6k + 5 that it's not on our list, or a composite number that is the product of an odd number of primes of the form 6k + 5. Those primes are added to the list and the process can go through another iteration.
The classic proof that there are infinitely many primes of the form 6k + 5 uses a similar process, but the algorithm is indifferent to whether the finite list has an odd or even number of primes. We take the product of the primes, multiply by 6 and then subtract 1.
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EXAMPLE
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To start things off, let's say 5 is the only prime of the form 6k + 5.
But 5 + 6 = 11, which is also a prime of that form. So our list is now {5, 11}. Since that has an even number of terms, we temporarily amend the list to {5, 5, 11}.
Then 5^2 * 11 + 6 = 281, which is also a prime of that form. Our list is now {5, 11, 281}.
Then 5 * 11 * 281 = 15461, which is prime. Our list is now {5, 11, 281, 15461}. Since that has an even number of terms, we temporarily amend the list to {5, 5, 11, 281, 15461}.
Then 5^2 * 11 * 281 * 15461 + 6 = 1194748781, which is prime. Our list is now {5, 11, 281, 15461, 1194748781}.
Then 5 * 11 * 281 * 15461 * 1194748781 + 6 = 285484928506498661 = 636653 * 448415272537, of which the former is a prime of the form 6k + 5 and the latter is not. Our list is now {5, 11, 281, 15461, 1194748781, 636653}. Since that has an even number of terms, we temporarily amend the list to {5, 5, 11, 281, 15461, 1194748781, 636653}
Then 5^2 * 11 * 281 * 15461 * 1194748781 * 636653 + 6 = 908774180942239441008581 = 41 * 101 * 4007847353 * 54756991297, of which only the last factor is not of the form 6k + 5.
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