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What Is an "Almost Prime" Number?

posted by Fnord666 on Monday November 05 2018, @02:29PM   
from the it-means-it's-composite dept.

upstart writes:

Submitted via IRC for Bytram

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.

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Original Submission

• If the IAU ran Mathematics If the IAU ran Mathematics (Score: 3, Funny) by theluggage on Monday November 05 2018, @07:42PM (1 child)

  by theluggage (1797) on Monday November 05 2018, @07:42PM (#758160)

  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.

  ...but has it cleared its own orbit?

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• Re:If the IAU ran Mathematics (Score: 0) by Anonymous Coward on Monday November 05 2018, @08:10PM

  by Anonymous Coward on Monday November 05 2018, @08:10PM (#758170)

  ...but has it cleared its own orbit?

  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.

  Parent