Post by: chiralSPO on 28/04/2022 16:41:40
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 throught of geometrically: if you have n things (let's say dots), then the only "rectangular" array of those n dots must be 1xn or nx1.
For example, in the image below, we can see that 3 dots can only be arranged in a rectangular array of 1x3 or 3x1. If we try making a 2x2, there is an incomplete edge. On the other hand, 4 is not prime because 2x2 works.
Screen Shot 2022-04-28 at 11.29.38 AM.png (85.49 kB . 990x1302 - viewed 7832 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 1x1 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 1x1 for 1. Only 1x2 or 2x1 for 2. Only 1x3 or 3x1 for 3. etc. And really, there is no geometrical difference between a Only 1x2 and a 2x1 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 1x4 (4x1) or 2x2. One can arrange 24 dots as 1x24 (24x1) or 2x12 (12x2) or 3x8 (8x3) or 4x6 (6x4).
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?
Post by: Eternal Student on 28/04/2022 18:22:32
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.
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Best Wishes.
Post by: Petrochemicals on 28/04/2022 18:27:27
I think the truth of it is would these primes be utilised in some thing like encryption.
Post by: evan_au on 29/04/2022 01:19:43
Quote from: Eternal Student
SPOILER: HIDEThat's cool!
How did you do it?
Post by: Eternal Student on 29/04/2022 03:00:28
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
Best Wishes.
Post by: evan_au on 29/04/2022 10:24:37
Spoiler: show
Post by: chiralSPO on 29/04/2022 19:19:50
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?
Post by: alancalverd on 29/04/2022 23:03:03
Except 2
Which is odd, because it's even.
Post by: Eternal Student on 30/04/2022 00:19:02
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
Best Wishes.
Post by: evan_au on 01/05/2022 04:11:50
- These are important in telecommunications and encryption schemes
Post by: chiralSPO on 01/05/2022 15:08:23
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.
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Post by: Halc on 01/05/2022 16:15:12
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.
Post by: Eternal Student on 01/05/2022 17:01:39
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.
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Best Wishes.
Post by: chiralSPO on 01/05/2022 18:02:44
Post by: Eternal Student on 02/05/2022 16:35:55
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.