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What Is an "Almost Prime" Number?
When I saw a math paper with the phrase "almost prime" in the title, I thought it sounded pretty funny. It reminded me of the joke about how you can't be a little bit pregnant. On further thought, though, it seems like someone whose pregnancy is 6 weeks along and who hasn't yet noticed a missed period is meaningfully less pregnant that someone rounding the bend at 39 weeks who can balance a dinner plate on their belly. Perhaps "almost prime" could make sense too.
A number is prime if its only factors are 1 and itself. By convention, the number 1 is not considered to be prime, so the primes start 2, 3, 5, 7, 11, and so on. Hence, a prime number has one prime factor. A number with two prime factors, like 4 (where the two factors are both 2) or 6 (2×3) is definitely less prime than a prime number, but it kind of seems more prime than 8 or 30, both of which have three prime factors (2×2×2 and 2×3×5, respectively). The notion of almost primes is a way of quantifying how close a number is to being prime.
(Score: 2) by vux984 on Tuesday November 06 2018, @10:32PM
Agreed. And thanks for the contribution of the ring theoretic angle!
Heh, pretty much exactly what I said in a different sub-thread :)
"The main reason 1 isn't considered prime is due to the fact that unique prime factorizations* are really useful. If 1 is allowed to be prime, then each positive number > 1 would not have a unique prime factorization; since you could repeat the 1 factor any number of times.
And yes, that does come from the fact that 1 is the multiplicative identity; so we exclude 1.
We could have just defined unique prime factorizations to only include prime numbers greater than 1; but since factorization is the most important application of primes, including 1 in the set, and then excluding it every time you used the set doesn't make a lot of sense.
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