[chiralSPO (OP)]

Got in a spirited debate the other evening (no pun intended), about whether the numbers 1 and 2 are prime.

The definition, as I know it from schooling, is that a prime number is "any number that is only divisible by itself and 1." ie if you can divide the number by any whole number that is not itself or 1, there must be a remainder or fractional component.

This can also be through of geometrically: if you have n things (let's say dots), then the only "rectangular" array of those n dots must be 1×n or n×1.

For example, in the image below, we can see that 3 dots can only be arranged in a rectangular array of 1×3 or 3×1. If we try making a 2×2, there is an incomplete edge. On the other hand, 4 is not prime because 2×2 works.

Screen Shot 2022-04-28 at 11.29.38 AM.png (85.49 kB . 990x1302 - viewed 2374 times)

1 is often left out from the primes because it has only one factor, itself (or 1). So it cannot be formed from an array of 1 by itself. Maybe this is a straw man argument (if so, please put me right). But this seems specious. A 1×1 array is still an array.

It seems to me that 2 should be treated the same as 1 because there isn't a choice among proportions of rectangular arrays. Only 1×1 for 1. Only 1×2 or 2×1 for 2. Only 1×3 or 3×1 for 3. etc. And really, there is no geometrical difference between a Only 1×2 and a 2×1 rectangle. The orientation is arbitrary (these aren't matrices).

Only when we get to non-prime numbers is there a choice of multiple types of rectangular arrays. One can arrange 4 dots as 1×4 (4×1) or 2×2. One can arrange 24 dots as 1×24 (24×1) or 2×12 (12×2) or 3×8 (8×3) or 4×6 (6×4).

There are many options for how to arrange 3 or more dots (only some of which. are rectangular arrays). There is only one option for 2 dots, and only 1 option for 1 dot.

So the real question I have is: if we count 2 as prime, should we not also count 1? Or, does the reason we don't count 1 also extend to 2?

[Eternal Student]

Hi.    Fantastic diagrams.   Great that you're making an effort to engage the audience with some Mathematics etc.
I wish you well.

I'll hide everything else under a spoiler because it's a bit dull and might prevent others from making their comments.

Spoiler: show

   The main reason for not counting 1 as a prime number is that most of the results we have about prime numbers, or more generally about whole numbers, won't work if you tried to state them as they are now and continued to use the term "prime number" in that statement of the result.   The decision not to include 1 as a prime number wasn't really done because of some elaborate definition or way of identifying what the primes are supposed to be.   I don't think following a pattern that emerges from dots had a lot to do with it.    Instead, it was done because it's not all that useful to have 1 included in the set.

    Another way to say this is that there's no reason you couldn't include 1 as a prime number if you want to.   You go right ahead and do that.   You don't even need to make up a good reason like drawing an arrangement of dots.   For whatever reason, you can put the number 1 into the primes if you like.   The only change that will result is that mathematicians will stop quoting their results by referring to "prime numbers" .  Instead they will identify a slightly different set of numbers, let's call them "Q-rimes" and their results will be stated with respect to that.   The Q-rimes will naturally be your Primes excluding the number 1.  So, the only thing that will have happend is that you will have changed the name we apply to describe what is currently called the prime numbers.

    I suppose to finish this I should give at least one example of a result that is useful and easily stated with reference to prime numbers (with 1 excluded but not if 1 is included).

     The fundamental theorem of arithmetic
Every counting number can be written as a product of prime numbers each raised to an appropriate (Natural number) power.   Furthermore, that respresentation is unique up to changing the order in which you perform the multiplication.

Example:     40  =  2³ x  5
   If you try to write 40 as some other product of primes, let's say you allow yourself to use three prime numbers   p, q, r    such that   40  = p^a x q^b x r^c     for some exponents  a,b,c   then you find that you can't,  there's no solution for that.     The fundamental theorem of arithmetic holds.
    However, if you allowed 1 to be a prime number then you can.....   One solution is to set p = 2,   q = 5, r = 1    and a= 3, b = 1, c= 2 .  That will be another representation of the number 40 as a product of primes:    40  =  2³ x  5  x  1²     and so the fundamental theorem of arithmetic doesn't hold.

Is 2 really a prime number?

    Actually 2 is another number that has very unusual properties even though it is prime.   It is often very useful and desirable for mathematicians to consider a subset of primes that doesn't include 2.   They call this set the "odd primes" and several theorems are stated with reference to  "odd primes" instead of just "the primes".  Alternative terms exist for this set and it's quite common not to bother naming the subset and just write a result as holding  "for all primes, p > 2".
   So 2 is a prime number but its certainly not typical of primes and there is a similar set, the odd primes, where you do just exclude it.

Best Wishes.

[Petrochemicals]

Possibly because it is not divisible, as in seperatable into multiple pieces. One thing I do know. Others are questionable, 2 is a prime but this is just binary, 3 is a prime and a triagonal number but again this only owes to the fact that its largest components are 1.

I think the truth of it is would these primes be utilised in some thing like encryption.

[evan_au]

SPOILER: HIDE

That's cool!
How did you do it?

[Eternal Student]

Hi.
    How did I add a spoiler?

Either:
     1.  Just use the tools along the top   (Office software would have called this the ribbon or tools bar)  and find the button marked    Sp    .

     2.   Just write your stuff between Spoiler tags that you added yourself.   You'll see what I mean if you try option 1 and the spoiler tags are added for you automatically.

Spoiler: show

   {spoiler}    and     {/spoiler}    except that you must use square brackets. 

Best Wishes.

[evan_au]

Spoiler: show

Spoiler Test
Thanks!

[chiralSPO (OP)]

Thanks, Eternal Student! That does make sense to me that the classification should be useful somehow. The phrase "odd primes" seems a clever way to make that distinction.

I came across another wrinkle: what about –1? It has exactly two factors, and can only be the product of 1 and itself (1×–1 or –1×1).

I think the answer here (and this is supported by the geometric interpretation), is that primes must be positive, or only positive factors can be considered. Otherwise –1 would be the only prime number as all others would have two pairs of ± factors (ie 7 could be decomposed into 1×7 and –1×–7, and all other positive primes would similarly have two positive and two negative factors, while –5 could be decomposed into 1×–5 and –1×5, and all other negative primes would similarly have two pairs of one positive/one negative factor).

Thoughts?

[alancalverd]

All primes are odd
Except 2
Which is odd, because it's even.

[Eternal Student]

Hi.

I came across another wrinkle: what about –1?

    Your general arguments after this are reasonable.   However, I think it is again just a matter of simplicity and having a set of numbers that are useful for something.   It is possible and useful to confine your attention to what people might call the counting numbers or the Natural Numbers, so we do.   That doesn't mean that mathematicians have never considered generalising the idea of prime numbers and investigating properties like prime factorisation in a structure bigger or more abstract than just the positive counting numbers - they certainly have.

   There is already some terminology you could use to describe a set of things that behave like prime numbers but apply to a much more generalised set of objects than just the Natural numbers.   These things are called "prime elements" and the parent algebraic structure is known as a "Ring".   You seem to be interested in the Ring which is the Integers (positive and negative Naturals with 0,  under  conventional binary operations of + and x).

   See   Wikipedia entry:   Prime elements, if you're interested.   https://en.wikipedia.org/wiki/Prime_element
However, you should note that they exclude "units" which would  include -1  in the ring of Integers,   i.e.   they would directly exclude both  +1  or  -1   from the prime elements in the ring of Integers.  (For what reason?   Similar to excluding 1 from the primes,  it makes it much easier to state an equivalent unique factorisation theorem for the ring of Integers).

    Here's a quick question or puzzle, just for fun.   It relates to the idea you mentioned earlier of eliminating the number 2 from the prime numbers.   You also seemed keen to extend beyond the positive numbers and consider negative numbers but you really don't have to stop there - you can consider Complex integers.
    The Complex Integers or "Gaussian Integers"   are the  Complex number equivalents of integers.   Specifically, the Gaussian integers are the set of all complex numbers of the form   a+bi   where  a and b are integers.
    Just like in ordinary arithmetic with Natural numbers, a prime  (or prime element) of the Gaussian integers is a Gaussian integer,  p,  that is irreducible or cannot be factorised.   Specifically,   if we have  p =  q × r   (where × is just ordinary multiplication of the complex integers q and r)  then  at least one of  q or r must be a unit element.     A  "unit"  is any complex number that lies on a unit circle around 0,  so the only  units in the  Gaussian integers are    +1, -1,  +i, -i.
   The number 2   is a prime in the ordinary integers.   Is it still a prime in the Gaussian integers?   To say that another way, can you factorise the number 2 in the Gaussian integers?

Spoiler: show

  2  =   (1+i) (1-i)   = the product of two Gaussian integers, neither of which are unit elements.   So 2 is not prime in the Gaussian integers.
  As it happens, the Gaussian integers do form a Unique Factorisation Domain.   This means there is a set of prime elements often called Gaussian Primes, all the Gaussian integers can be written as a product of those prime elements and, as always, that factorisation is unique.   However, its prime elements are quite different to the prime numbers of the ordinary Integers.

Best Wishes.

[evan_au]

There are also prime polynomials (if you ignore imaginary zeroes).
- These are important in telecommunications and encryption schemes

[chiralSPO (OP)]

Thanks for introducing me to larger sets (groups?) of "prime" things.

I had one more thought of a geometrical reasoning for defining the odd primes as a reasonable set:

"prime numbers" can be defined as regular polygons with n sides that can only be divided into n identical portions by connecting vertices to the center. Non-primes have more than one way to connect verteces to the center to form congruent subsections.


Screen Shot 2022-05-01 at 9.58.01 AM.png (92.41 kB . 1112x912 - viewed 1265 times)

[Halc]

Your definition of a prime (in the OP) includes 1, but the definition asks for 'divisible by 1 and itself' and 1 is not a divisor of 1 in addition to the first 1. That's pretty shaky, but every integer > 1 can be factored into exactly one set of primes, and this would not be true if 1 was a prime. So 6 could be factored into 2,3 or 1,1,1,2,3 and the list goes on.

As for your geometry example just now, besides leaving 2 off the list, your definition as worded doesn't work as worded.
"regular polygons with n sides that can only be divided into n identical portions by connecting vertices to the center"
Take your hexagon. There is only one figure below it that divides it into 6 portions. The others are not divided into 6 so don't count. You might say "regular polygons with n sides that can only be divided into identical portions by connecting vertices to one common additional point".  The mention of the center is unnecessary since I don't see a way to do it without picking that point.  The "one common" part seems necessary since I can happily divide any regular polygon into many identical portions. The pentagon for instance can be slice up into 20 identical shapes, but only with the addition of more than one new point where the lines intersect.

Another possible wording:
"Any n>4 is not prime if a regular n-sided polygon can be divided into m identical portions, with 2 < m < n". Given that wording, my dividing the pentagon into 20 shapes doesn't disqualify it as being prime, and even counting mirror images as identical doesn't trip up the definition.

[Eternal Student]

Hi.

   I have to agree with the spirit of what @Halc   is saying.   There's a lot in your ( @chiralSPO  ) polygon definition that may need some clarification or just doesn't seem to work.

   Maybe you were trying to say:   There is only one way to divide the p - sided regular polygons into smaller identical portions?    However that doesn't hold true.  Even the simplest prime sided polygon, the equilateral triangle can be divided up into 3 portions as you have shown or 6 identical portions as shown below (admittedly some portions are reflections or equivalently rotations in the 3rd dimension).

Best Wishes.


triangle.png (9.38 kB . 1171x633 - viewed 1138 times)


Best Wishes.

[chiralSPO (OP)]

Indeed. I was careless. I meant to divide each n-gon into m equal portions by connecting vertices to the center.

[Eternal Student]

Hi.

   I think I can see what you meant @chiralSPO .
Start with an n-gon and always divide it up by drawing some lines from some of the vertices to the centre.   The aim is to divide the n-gon into identical pieces.    If n = prime number then there should only be one way to do that,  join all the vertices to the centre.

    That sounds reasonable and a rough or sketch proof of that result doesn't seem too difficult.   If you're interested we could write it down.     However, as a method of defining the prime numbers I'm not sure it's better or faster than the existing definition(s).
     As Halc mentioned it also completely leaves out the number 2 but perhaps that was your original aim.

Best Wishes.