Is 1 Prime, and Does It Matter? • Hacker News Board

Re: Is 1 Prime, and Does It Matter?

#41

Post by alganet » Mon, Apr 21, 2025, 9:01 PM UTC

"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 will become outliers and a distribution will reveal itself along the 1, 5, 7 and 11 diagonals).

Anyways, "only divisible by itself and 1" is a darn elegant definition.

Re: Is 1 Prime, and Does It Matter?

#42

Post by dullcrisp » Mon, Apr 21, 2025, 9:01 PM UTC

ks2048 wrote: post #35

Can we declare 2 composite? Kind of annoying to have an even number in there.

It’s composite in the Gaussian integers, maybe that helps.

Re: Is 1 Prime, and Does It Matter?

#43

Post by stouset » Mon, Apr 21, 2025, 9:02 PM UTC

Maxatar wrote: post #38

gerdesj wrote: post #27

Earlier quoted context omitted.

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 ...

Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers.

Do you have a link to where I could learn more about this?

Re: Is 1 Prime, and Does It Matter?

#44

Post by IshKebab » Mon, Apr 21, 2025, 9:07 PM UTC

fpoling wrote: post #33

dullcrisp wrote: post #5

This is like a "do arrays start at 0 or 1" question, except as they mention, algebraic number theory pretty much settles it. Whether 0 is a natural number though is still open for bikeshedding.

I always thought that 0-based indexes were superior until few years ago I needed to deal with Fortran code and I realized that 1-based arrays allowed to use 0 as a non-existing index or sentinel, not size_t(-1) hack as found in C/C++. Like the article explains, depending on the domain one or the other convention can be advantageous. And then C/C++ compilers are subtly inconsistent. If 0 is valid index, then null shou…

No, this article makes the case for 0-based indexing. Let's ignore the reality that computer fundamentally use 0-based indexes... The article says 1 is not prime because maths gets more awkward if it is.

In the same way we index from 0 because indexing gets way more awkward if we index from 1.

In-band sentinels are both quite rare, and also equally convenient with -1 or 0. In fact I would say -1 is a bit more elegant because sometimes you need multiple sentinel values and then you can easily use -2 (what are you going to use 0 and 1 and then index from 2?).

The more common operations are things like indexing into flattened multidimensional arrays, or dealing with intervals, which are both way more elegant with 0-based indexing.

0 is a valid index into an array. It's even a valid index into global memory in some environments. Not mapping memory to address 0 is completely trivial. I'm not sure what non-trivial complications you're thinking of.

Re: Is 1 Prime, and Does It Matter?

#45

Post by gerdesj » Mon, Apr 21, 2025, 9:07 PM UTC

stouset wrote: post #43

Maxatar wrote: post #38

Earlier quoted context omitted.

Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers.

Do you have a link to where I could learn more about this?

You might start here: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

That's the GEB I mentioned above.

Re: Is 1 Prime, and Does It Matter?

#46

Post by rtkwe » Mon, Apr 21, 2025, 9:09 PM UTC

feoren wrote: post #39

All models are wrong, but some models are useful. It's not useful to consider 1 prime, so we don't. You're free to invent a new model of math where 1 is prime and see where it takes you; nobody will be offended. This happens all the time: "but what if we could take the square root of a negative number? What then?", etc. 99% of the time, this leads to a theory that is provably inconsistent and therefore useless. Out o…

I've always wondered what actually breaks if 1 is prime or conversely what defining 1 as not prime gives us. Got just far enough into my math degree before switching to CompSci to stay of of universities the rest of my life to want to know.

Re: Is 1 Prime, and Does It Matter?

#47

Post by overboard2 » Mon, Apr 21, 2025, 9:14 PM UTC

Maxatar wrote: post #38

gerdesj wrote: post #27

Earlier quoted context omitted.

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 ...

Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers.

What do you mean by "not actually"?

Edit: do you mean literally impossible?

Re: Is 1 Prime, and Does It Matter?

#48

Post by murderfs » Mon, Apr 21, 2025, 9:19 PM UTC

JadeNB wrote: post #20

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".

> 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…

A good example of this is the natural numbers. Algebraists usually consider zero to be a natural number because otherwise, it's not a monoid and set theorists want zero because it's the size of the empty set. My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why.

Re: Is 1 Prime, and Does It Matter?

#49

Post by pinkmuffinere » Mon, Apr 21, 2025, 9:22 PM UTC

gerdesj wrote: post #45

stouset wrote: post #43

Earlier quoted context omitted.

Do you have a link to where I could learn more about this?

You might start here: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach That's the GEB I mentioned above.

I know this is a great book, it’s been on my to-read list for about 5 years. But I never get to it. Is there not another (shorter) discussion I could read on this? Even an academic paper would be acceptable.

Re: Is 1 Prime, and Does It Matter?

#50

Post by jordigh » Mon, Apr 21, 2025, 9:30 PM UTC

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-...