[deleted]
Is 1 Prime, and Does It Matter?
21–30 of 162 posts
Re: Is 1 Prime, and Does It Matter?
#22I'm sure it depends on the definition of prime. I've always been partial to "Any integer with exactly 2 divisors". Short, simple, and it excludes 1 and negative numbers.
> I'm sure it depends on the definition of prime. I've always been partial to "Any integer with exactly 2 divisors". Short, simple, and it excludes 1 and negative numbers.
Depending on your definition of divisor, it excludes everything except 1 and -1, whose two integer divisors are 1 and -1. But then, if you specify that "divisor" means "positive integer divisor", it no longer automatically excludes the negative numbers, since the two positive integer divisors of -2 are 1 and 2. (Incidentally, plenty of algebraists, myself included, are perfectly comfortable with including -2 as a prime.)
Re: Is 1 Prime, and Does It Matter?
#23If 1 is prime, then the fundamental theorem of arithmetic goes from "every positive integer can be written as a product* of primes in one and only one way" to "every positive integer can be written as a product of primes greater than 1 in one and only one way". Doesn't quite have the same ring to it. So just from an aesthetic perspective, no I'd rather 1 isn't a prime number. * empty products being 1 of course
It seems a little inconvenient to require acceptance that empty products equal 1, since that is also slightly subtle and deserving of its own explanation of mathematical terminology.
Of course, I generally hear the fundamental theorem of arithmetic phrased as “every integer greater than one…” which is making its own little special case for the number 1.
Re: Is 1 Prime, and Does It Matter?
#24I think we’ll need to wait for an answer to if there is a prime number generating function. At that time we can determine if 1 is prime. If it’s found that Eratosthenes’ sieve is the only prime generating function then we have our answer.
We'd have our answer in what way?
If you treat one as prime number when running the sieve algorithm, one is the only prime number that remains after you have removes all its multiples from the list of candidate numbers.
Re: Is 1 Prime, and Does It Matter?
#25Earlier quoted context omitted.
Isn't "every positive integer can be written as a product of primes greater than 1 in one and only one way" incorrect? A prime number is a only product of itself * 1, isn't it?
Mathematicians generally feel that a single number qualifies as a "product of 1 number." So 7 can be written as just 7 which is still considered a product of prime(s). This is purely a convention thing to make it so theorems can be stated more succinctly, as with not counting 1 as prime.
Ah, OK, thank you.
Re: Is 1 Prime, and Does It Matter?
#26Earlier quoted context omitted.
Isn't "every positive integer can be written as a product of primes greater than 1 in one and only one way" incorrect? A prime number is a only product of itself * 1, isn't it?
1 is not greater than 1, and a product of one prime is still a product of primes
Yeah, I didn't understand you can have a product of a single number.
Re: Is 1 Prime, and Does It Matter?
#27One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".
Yes, it's more of a convention where we assume language like "...ignoring the trivial case of 1 being an obvious factor of every integer." It's not interesting or meaningful, so we ignore it for most cases.
I'm no expert but:
"...ignoring the trivial case of 1 being an obvious factor of every integer."
I remember quite a big chunk of GEB formally defining how integers are really not trivial! The main problem seems to be is that you soon end up with circular reasoning if you are not razor sharp with your definitions. That's just in an explainer book 8)
Then you have to define what factor means ...
Re: Is 1 Prime, and Does It Matter?
#28One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".
> One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1". This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently e…
That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth.
That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable!
Re: Is 1 Prime, and Does It Matter?
#29Since 2 is prime 1, wouldn't it be more symmetric if 1 was prime 2?
What makes you think two is prime? Not everyone would agree with that statement as the artical points out.
Re: Is 1 Prime, and Does It Matter?
#30[dead]