Co-primes

I came across this post about ABC conjecture. If you haven’t read it, you should it’s good writing.

Now somewhere at the start of the 2nd paragraph he states that if a and b are relatively prime(i.e, they both are not divisible by a common number n>1), then the sum c = a + b is also relatively prime to both a and b.

Now, my first reaction was obviously and skip it over, but then i asked myself why, and after some 30-40 mins of mental wrangling* proved to myself it is the case.
Here’s how:
Initially, just for surety, i asked is it true? So, i tried out some basic prime numbers, started with 2 and 3 went on to 2,3,5,8,13,21 etc.. and stopped and asked why?

I started of with wondering trying to figure out how the fact that we added those two numbers said something about their factors(i.e: a multiplicative operation). Then after some minutes of brain mashing, it hit me.

Ok, what exactly is relatively prime? it is not having a common clean divisor. now what does a clean divisor mean? It means the ability to express a number as a repeated addition of another (absolute??)prime number. Once i got that, it was straight forward algebra to work out the proof.

Relatively prime means, there is no number > 1 that can form all(in this case, both) the given numbers by repetitive addition.

So if you add relatively prime numbers, the resulting sum cannot but help being relatively prime to the added numbers. Because if it did it would mean that the sum has prime factors that is in common with the individual numbers prime factors, which is impossible since the individual numbers are relatively prime and prime numbers are non-factorizable by definition.

Ok, so far titling this post as category theory is outrageous, and far-fetching. I’ll explain my reasoning why, i was reminded of category theory, but am afraid it still might be far-fetched. Anyways, once i saw the relationship between multiplication and addition, my first recall from memory was groups. Numbers + operators(in this case addition and multiplication on whole natural numbers).

I’ll go on and try to explain the abc-conjecture now. or rather re-explain it. you really should read the original article(linked at the top).

To get to the conjecture, we need to define one more set/concept/number called Radical of a,b,c.
The way you find the Radical is you find all the prime factors of the given numbers(a,b,c with c=a+b, in this case) and then eliminate duplicates and finally multiply the rest.

To use the same operations, i used above, it is like the splitting the numbers into their prime factors.(i.e: smallest possible numbers, that can make up the given number by repetitive addition, and cannot themselves be made up by repetitive addition of other numbers).
Expressed that way, without the multiplication operator, a number like 24 can be expressed two ways.
i.e: 24 = 2 + 2+ 2+2+2+2+2+2 + 2 + 2 +2 +2 or
24 = 3 + 3 + 3 + 3 +3 +3 + 3 +3

So both 2 and 3 are prime factors of 24.

So, when you are given a set of numbers, you find all of it’s prime factors and then remove the duplicates(in the case of our a,b and c) there wouldn’t be.(relatively prime remember?) and then multiply them together.

Or rather repeatedly add the first factor as many times as the second and then the result as many times as the third and so on..

The abc-conjecture states that for any number h>1 there’s a finite set {a,b,c} that satisfies the relationship R raised to the power of h < c.

PS: Man it’s a pain to replace multiplication by repetitive addition. I tried to use only addition operation throughout that post and gave up towards the end when i came across exponentiation.

PPS: Also, i know i originally promised to write more about applications of category theory, but got distracted with this paper getting popular :-). I will try to get back the original quest next week. Cya all.

*– Rather rewarding experience in terms of the pleasure center stimulation it has resulted in.

Combinatorial Species– my comprehension.

Disclaimer: This post is not accurate, nor am i a professional mathematician(not even amateur) . Feel free to correct my misunderstandings in the comments.

I was trying to understand Combinatorial species(https://en.wikipedia.org/wiki/Combinatorial_species). came across somebody’s profile at the company indix .  But it had references to functors, categories and i went off reading Category theory here).  Couple of hours later, i still don’t have an understanding of Combinatorial Species, but I have a (slightly) better (than my previous ) grasp of Category theory.  Let me try to summarise that.

 

Back in High school I think 11th grade, we had a small cute, and fun chapter called modern algebra. It’s one of those chapters, that left me with wanting for more. Very small simple chapter, but it was beautiful.  It defined groups, sets,operators, types and properties of a group etc..

Guess what? it turns out Category theory is just one level of abstraction higher.  While as i remember that ‘modern algebra’ dealt with sets (of numbers were the examples we used) and operators, Category theory deals with objects and arrows.  Note the vagueness of the definition of objects and arrows is on purpose.

If the object is a set and the arrows are operators we get a group.

If the objects are data types and the arrows are functions we get type system/theory from theoretical comp. science.

<More examples>

I am trying to think what if objects are distributions and the arrows are transformations? I can’t think of what area that would be, but would be surprised if it has not been studied formally.

Anyway, the fact that objects and arrows are not restricted does not mean there are no rules. there are a some rules for considering a <i don’t want to use the term set/group> collection of objects and arrows(hereafter called collection ) as a category as follows:

Ah, i missed one more thing so far . arrows are not just arrows they are mathematical morphisms/maps  defined between objects. at this point am thinking why this sounds very much like set theory and not really any different from sets and functions/mapping functions.   why the hell would i need yet another set of terminologies to master*

Any way, here are the rules for a collection to be considered a category.

1. the morphisms must be composable to achieve a associative relationships among the objects.

i’ll use u,v,w for morphisms and a,b,c, for objects.

i.e  if u: maps a to b, v: maps b to c, then u compositing v: maps a to c.

 

 

2. That composition function between morphisms must be associative itself.

i.e:  (u composits v) composits w = u composits (v composits w)

 

3. There’s an identity morphism I such that  for any u : maps a to b, I composits u = u composits I  = u.

 

*– Will get back to that later i need to read a lot more on the background, history, and some papers to understand why it is useful. but the wikipedia explanation is that it abstracts and unifies concepts many different branches of mathematics. While it sounds true, it’s rather vague(insert rant about perils of democratic editing/article writing), i want to see some actual theorems in Category theory being applied to some math area and helping improve it. Will write another post/update this once am done with that.

UPDATE: 11-Sep-2012: Seems i was rash commenting about utility of category theory. This link suggests a new revolutionary proof on prime numbers uses some concepts from category theory.