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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, Insightful) by Anonymous Coward on Monday November 05 2018, @03:03PM (9 children)
1 isn't a non-prime purely by convention, it's a byproduct of being the multiplicative identity. A prime is essentially any number that only has 2 whole number factors 1 and itself. 1 has factors of the form 1^x, meaning that it fails the definition. And because it fails the definition they had to put a caveat that the paper doesn't apply to 1 so often that they went the extra step of specifying that it's not prime even though it probably shouldn't have been in the first place.
(Score: 2) by DannyB on Monday November 05 2018, @06:01PM (1 child)
1 also has factors of the form x0.
The lower I set my standards the more accomplishments I have.
(Score: 2) by vux984 on Monday November 05 2018, @06:41PM
so does every other number. :)
(Score: 2) by wonkey_monkey on Monday November 05 2018, @07:15PM (3 children)
We could avoid all this "except 1" nonsense if we just went with "prime numbers are integers with exactly two divisors."
systemd is Roko's Basilisk
(Score: 2) by wonkey_monkey on Monday November 05 2018, @07:18PM (2 children)
Postive integers, that is.
systemd is Roko's Basilisk
(Score: 0) by Anonymous Coward on Monday November 05 2018, @08:02PM (1 child)
To be clear: "Prime numbers are positive integers with exactly two (i.e., one pair of) positive integer divisors." Here "divisor" is also assumed to be the more restrictive definition of a number which divides into another without remainder, rather than simply a number that divides into another.
Not sure this is a simpler definition of "prime number."
(Score: 2) by wonkey_monkey on Tuesday November 06 2018, @05:41PM
It's simpler because it doesn't need an exception to be specified.
systemd is Roko's Basilisk
(Score: 2) by vux984 on Monday November 05 2018, @07:23PM
You are right but you wrote that really awkwardly.
"A prime is essentially any number that only has 2 whole number factors 1 and itself."
So... 1 has 1 as a factor. Check. 1 has itself as a factor. Check. There are no other factors: Check. So it's prime?
Or did you mean that the "2 whole numbers" had to be 2 different whole numbers? because that could be read as just enumerating the two criteria of its factors. And the use of 'and' in '1 and itself ' as a logical operator does not imply the two operands not be identical. Also you forgot positive, so now -1 is maybe prime too. :p
"1 has factors of the form 1^x, meaning that it fails the definition"
Why? 1^x = 1. Your "essential" definition of prime clearly says 1 is allowed. The fact that you found out how to write 1 a 2nd way doesn't change that.
I mean 17 has factors of the form ((17+x)-x) but that doesn't make 17 not prime, right?
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.
Although 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 everytime you used the set doesn't make a lot of sense.
* Fundamental Theorem of Arithmetic states that each positive number greater than 1 ** has a unique prime factorization.
** See how many other theorems and proofs also have to explicitly exclude 1 from the set of positive numbers! We should have just excluded 1 from the set of positive whole numbers too, right? :) Except it would be really weird if 1 were excluded from positive numbers when dealing with rational and real numbers, and inconsistent to include it in positive reals but not positive integers...
Meanwhile there aren't really any cases where having 1 in the set of primes gives us anything especially useful, and no inconsistencies arise by leaving it out.
(Score: 3, Funny) by theluggage on Monday November 05 2018, @07:42PM (1 child)
...but has it cleared its own orbit?
(Score: 0) by Anonymous Coward on Monday November 05 2018, @08:10PM
Yes. Proof? Graph any function f(x)=b^x where b>0 is not 1. The function will blow up toward infinity one way or another (depending on whether b is less than 1 or greater than 1). For an arbitrary number of factors (e.g., b^x * c^y * d^z *...., where b, c, d, etc. are all different and not 1), the function will blow up as the exponents increase. This simulates any possible composite number with factors other than 1.
Now graph f(x) = 1^x. It's a line. Any other number has limits that are infinitely far away from the line y=1. We must therefore conclude that all the other integers are afraid of 1 and run away from it over time. Hence, the number 1 will definitely clear its "factor orbit" as we go toward infinity, just like any good multiplicative identity.