Point of Clarification: I am not claiming that σ(n) cannot contain a multiple of 2. IF I was, it would mean I am arguing that σ(n) cannot be even, but that is not the case given that the identity for both primes and perfect numbers in the sigma function are even, in other words primes (aside from 2) and perfect numbers always result in an even sigma sum. I am stating that the sigma sum σ(n) cannot be the even positive integer 2. If you're not convinced, grab a pen and piece of paper and start calculating σ(n) for all positive integers starting with the smallest positive integer of 1 and let me when you think you'll find a positive integer whose sum of divisors will add up to a total of 2. Go ahead...I'll wait ;)...shouldn't take long to understand what I'm claiming. (This post was inspired by a comment that claimed I was trying to say σ(n) can't contain a multiple of 2. (I think I know what my next video post will be about - a spotlight on the sum-of-divisors function)
Something worth pondering might be that presenting something and claiming that it doesn't exist, is already prima facie evidence of it's existence! ;) Even 'infinity' exists! Wanna see it again; >>>> infinity. See?
In an abstract world without any constraints, you are correct. In fact σ(n) = 2 does exist but not for Positive Integers. Context is meaning. In this context, we are dealing with extremely strict constraints - those of Positive Integers. You have to abide by the rules of the world you create - that is the logic part in a logical conclusion. Otherwise everything in mathematics would be meaningless and everything would be both possible and impossible at the same time all the time - and that would make this all a waste of time.
@@abinitio_adinfinitum Ahh, the bugaboo is in the constraints! With sufficient 'focus', much can be highlighted and much more can be ignored. Depending where you place the boundaries. The First Law of Soul Dynamics says; "For every Perspective, there is an equal and opposite Perspective!" And so with the Perspectives of 'possibility' and 'impossibility'. All 'meaning', like art and beauty and everything else, exists in the concepts/Mind of the beholder. Time can only be 'wasted' from certain Perspectives, opposite for others and a wide spectrum between.
Point of Clarification: I am not claiming that σ(n) cannot contain a multiple of 2. IF I was, it would mean I am arguing that σ(n) cannot be even, but that is not the case given that the identity for both primes and perfect numbers in the sigma function are even, in other words primes (aside from 2) and perfect numbers always result in an even sigma sum.
I am stating that the sigma sum σ(n) cannot be the even positive integer 2. If you're not convinced, grab a pen and piece of paper and start calculating σ(n) for all positive integers starting with the smallest positive integer of 1 and let me when you think you'll find a positive integer whose sum of divisors will add up to a total of 2. Go ahead...I'll wait ;)...shouldn't take long to understand what I'm claiming.
(This post was inspired by a comment that claimed I was trying to say σ(n) can't contain a multiple of 2. (I think I know what my next video post will be about - a spotlight on the sum-of-divisors function)
Something worth pondering might be that presenting something and claiming that it doesn't exist, is already prima facie evidence of it's existence! ;)
Even 'infinity' exists! Wanna see it again; >>>> infinity.
See?
In an abstract world without any constraints, you are correct. In fact σ(n) = 2 does exist but not for Positive Integers. Context is meaning. In this context, we are dealing with extremely strict constraints - those of Positive Integers. You have to abide by the rules of the world you create - that is the logic part in a logical conclusion. Otherwise everything in mathematics would be meaningless and everything would be both possible and impossible at the same time all the time - and that would make this all a waste of time.
@@abinitio_adinfinitum Ahh, the bugaboo is in the constraints! With sufficient 'focus', much can be highlighted and much more can be ignored. Depending where you place the boundaries.
The First Law of Soul Dynamics says; "For every Perspective, there is an equal and opposite Perspective!"
And so with the Perspectives of 'possibility' and 'impossibility'.
All 'meaning', like art and beauty and everything else, exists in the concepts/Mind of the beholder.
Time can only be 'wasted' from certain Perspectives, opposite for others and a wide spectrum between.