Just a few notes to clarify Furlong's Postulate! A "twin prime" is usually taken to mean 2 prime numbers that are separated by 1 and only 1 even composite number, which must necessarily be some multiple of 6, except for the very first set of twin primes, namely, 3 and 5, which are separated by 1 and only 1 even composite number (4), which is manifestly not some multiple of 6. Just as I did before in the case of Bertrand's Postulate, we can easily verify Furlong's Postulate by "daisy chaining" up the number line: the 3rd twin prime pair (11, 13) is strictly between 7 and 2*7 = 14, the 4th twin prime pair (17, 19) is strictly between 13 and 2*13 = 26, the 5th twin prime pair (29, 31) is strictly between 19 and 2*19 = 38, the 7th twin prime pair (59, 61) is strictly between 31 and 2*31 = 62, the 10th twin prime pair (107, 109) is strictly between 61 and 2*61 = 122, the 15th twin prime pair (197, 199) is strictly between 109 and 2*109 = 218, and the 21st twin prime pair (347, 349) is strictly between 199 and 2*199 = 398, and so forth!
By the way, I have no earthly idea how anyone would go about trying to prove my postulate for all primes! It's not even known for sure whether there are an infinite number of twin primes! Almost every number theory nut worth their salt believes that there must be an infinite number of twin primes! Of course, if someone could prove that my postulate is true, somehow, then, since there are an infinite number of prime numbers, that would necessarily mean that there would have to be an infinite number of twin primes!
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