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Is 1 Prime, and Does It Matter?

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Re: Is 1 Prime, and Does It Matter?

#91

Post by croes »

rvba wrote: post #58
munchler wrote: post #18

Other good nerd-sniping math questions: 0^0 = 1? Yes, it’s simpler that way. 0! = 1? Yes, it’s simpler that way. 0/0 = ∞? No, it’s undefined. 0.9999… = 1? Yes, it’s just two ways of expressing the same number. 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.

If we try to define division by zero, shouldnt 0/0 be 1? Or even more abstract "every element on y". Which I think could sort of work

But that would mean (0/0) * 2 = 2 but (0/0) * (2/1) = (0 * 2) / (0 * 1) = 0/0 = 1

Re: Is 1 Prime, and Does It Matter?

#93

Post by chris37879 »

2OEH8eoCRo0 wrote: post #74

1 x 1 = 1 1 x 1 x 1 = 1 ... Not prime!

Depends on your definition of prime, by your reasoning, I could say 7 * 1 * 1 = 7, so it's not prime. Better to say a prime is any number with a set of divisors of length 2 including 1 and itself. If you want to exclude 1.

Re: Is 1 Prime, and Does It Matter?

#94

Post by tux3 »

JadeNB wrote: post #86
tux3 wrote: post #28

Earlier quoted context omitted.

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!

> That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable! "Wellspring of dryness" is quite a metaphor, and I take it from that metaphor that this outcome wouldn't much bother you. I'll put in a personal defense for set theory, but only an appeal to my personal taste, since I have no expert, and barely even an amateurish, knowledge of…

I should apologize if I came off too colorful, I only meant it as a friendly jab - but my bias is showing :)

Appreciate the defense of set theory, I can't find a problem with it!

Re: Is 1 Prime, and Does It Matter?

#95

Post by chongli »

JadeNB wrote: post #62
gus_massa wrote: post #36

Earlier quoted context omitted.

Many (most?) results are easier to write if you allow the empty set. For example: " The intersection of two sets is a set. "

> Many (most?) results are easier to write if you allow the empty set. For example: > "The intersection of two sets is a set." Many results in set theory, yes! (Or at least in elementary set theory. I'm not a set theorist by profession, so I can't speak to how often it arises in research-level set theory.) But, once one leaves set theory, the empty set can cause problems. For the first example that springs to mind, i…

I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty.

The same is true for any structure which posits the existence of some element. Of course it cannot be the empty set.

Re: Is 1 Prime, and Does It Matter?

#96

Post by alganet »

andriamanitra wrote: post #90
alganet wrote: post #41

"Only divisible by itself and 1" is a darn elegant definition. 1, 2 and 3 are kind of special to me. In prime distribution studies, I discovered that they are special. It gets easier for some things if you consider primes only higher or equal to 5. Explaining distribution gets easier, some proofs become more obvious if you do that (tiny example: draw a ulam-like spiral around the numbers of an analog clock. 2 and 3 w…

It's not entirely clear if that definition includes 1. On one hand 1 is certainly divisible by both itself and 1, but on the other hand they are the same number, so maybe it shouldn't count for "both", because the word "both" vaguely implies two distinct things. The usual "natural number with exactly two integer divisors" definition may not be as elegant but I think it is harder to misinterpret.

I never used the word "both" there.

But thanks anyway! I learned a thing.

Re: Is 1 Prime, and Does It Matter?

#97

Post by arcastroe »

jordigh wrote: post #50
JJMcJ wrote: post #3

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".

To be fair, 2 is also a very odd prime because it's even. So many theorems have to say, "for every odd prime..." https://math.stackexchange.com/questions/1177104/what-is-an-...

"2 is the only even prime number. Therefore, it's the oddest of them all!"

Re: Is 1 Prime, and Does It Matter?

#98

Post by sam_ezeh »

Maxatar wrote: post #88
sam_ezeh wrote: post #87

Earlier quoted context omitted.

>Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers This isn't correct. This is only true for first-order theories of the natural numbers using the axiom schema of induction. Second-order Peano arithmetic with the full axiom of induction has the natural numbers as its only model. This property is called "categoricity" and you ca…

This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation. You can adopt Henkin semantics to give the naturals an interpretation, which…

> So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation.

Can you explain what you mean here? Full semantics for second-order logic has a unique interpretation i.e. the standard natural numbers

Re: Is 1 Prime, and Does It Matter?

#99

Post by margalabargala »

munchler wrote: post #18

Other good nerd-sniping math questions: 0^0 = 1? Yes, it’s simpler that way. 0! = 1? Yes, it’s simpler that way. 0/0 = ∞? No, it’s undefined. 0.9999… = 1? Yes, it’s just two ways of expressing the same number. 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.

> 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.

The other ones, sure, but I'm not following this one.

Re: Is 1 Prime, and Does It Matter?

#100

Post by tshaddox »

feoren wrote: post #32
tshaddox wrote: post #23

Earlier quoted context omitted.

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.

>It seems a little inconvenient to require acceptance that empty products equal 1 Only the contrary: it is extremely inconvenient to not allow the product of an empty sequence of numbers to equal 1. The sum of an empty sequence is 0. The Baz of an empty sequence of numbers, for any monoid Baz, is the identity element of that monoid. Any other convention is going to be very painful and full of its own exceptions. Ther…

That’s not what I meant. I agree that the empty product being equal to 1 is reasonable.

I meant that it’s inconvenient to require engaging with that concept directly in the everyday definition of prime numbers.

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