For example, if we were to reduce the number 108 to its prime factors, it is:
2 • 2 • 3 • 3 • 3
If we were to include 1 as a prime factor, then every integer (including 1) would have an infinite number of prime factors.
For example, the number 108 “reduces” to
2 • 2 • 3 • 3 • 3 • 1 • 1 • 1• 1• 1• 1• 1• 1• 1 …
It literally carries on ad infinitum.
So, maybe a better definition of a prime number would be an integer that has only 1 prime factor? (The special case of 1 would have zero prime factors).
I know one is not prime. At least that’s what they told me in HS. But the definition I remember was something like, “A number is prime if it can be divided evenly only by itself and one.” Since one is evenly divisible only by itself and one, they had to have an extra rule to give one the boot: “PS, one is not a prime number.”
Totally bogus, IMO.
When you have to make a special rule to keep somebody out, you know you’re dealing with some kind of prejudice.
Pluto, on the other hand, is a totally different matter. If you’re going to let Pluto in, you’ve got to let every puny little hunk of rock in the Kuiper belt in as well - we’re going to have 20 million planets, pretty soon.
Plus, the solar system as a special kind of symmetry without Kuiper belt objects. Four small hard planets in the inner ring; four large gas giants on the outside, with the asteroid belt there to keep them apart. Pluto is little more than a round-ish shaped asteroid itself, that happens to be floating around periphery of the system.
One, on the other hand, totally deserves to be prime. I mean, it’s frikken “1”!
Along with what wolf_meister said, 1 used to be considered prime. But the convention now is to exclude 1 so that the prime factor decomposition of every composite number is unique.
No señor… uno, tres, cinco, siete, where’s the e in uno and cinco, eeeeh?
The problem with using numbers as part of an IQ test is that if you toss in things like “a series of prime numbers,” you assume that the test-taker knows what prime numbers are, which is a matter of knowledge and not brute intelligence. Add language to it and you’re shooting your own foot off.
One could, using “brute intelligence,” figure out the concept of prime numbers (without knowing that’s what they were called). (The same could be said for odd numbers, for that matter.)
If I gave the sequence 2, 3, 5, 7, 11, … and asked what comes next, someone who had never encountered the concept of prime numbers could still figure out the answer 13. But someone who was familiar with prime numbers would recognize this a lot more quickly.
I got the answer to the OP’s question right away, but that’s because I’d encountered that question, or one very much like it, before. I suspect that’s true of any “IQ Test Question”: it’s going to depend on knowledge or experience instead of just “brute intelligence,” because people who have encountered that question or that kind of question before are going to have an advantage.
Every finite field has q elements, where q=p[sup]n[/sup] for some prime p and some natural number n, and any two finite fields of the same order are isomorphic. We write F[sub]q[/sub] for the (unique up to isomorphism) field of order q.
When q is a power of 2 you run into weirdness. For example, quadratic forms over F[sub]q[/sub] are problematic – since 1+1=0, you have trouble symmetrizing them like you want because you can’t divide by 2. Many results hold for q a power of any prime except 2, and “odd prime” is a concise way of saying “any prime other than 2”.
One might say that part of the reason that odd primes are more special than, say, primes not divisible by 11 is that 2 is more special than 11. Not that there’s any formal theory of specialness that I know of, but, intuitively speaking, 2 is particularly exceedingly special, as far as numbers go.
