Why Number Theory is Hard (Audio Fix in Description)

Number theory is a field of mathematics that dates back to the earliest civilizations, but why is it so notoriously difficult? In essence, it boils down to addition and multiplication each cannibalizing the nice properties the other brings to the table.

Notes:

Scaling a vector by 0 always produces the 1-vector, corresponding to the fact that n^0 = 1 for all integers n.

#math #numbers #addition

[DontWatchWhileHigh]

The real problem is that we have way too many numbers.

[JohnnySacc]

good vid but the inception sound blew my ears out

[xiyition]

put a normaliser on your audio the loud sounds are rather jarring

[octopeople]

Factor space can additionally represent all positive rational numbers, if you allow negative components in the vectors

[minerharry]

This is *fascinating*, awesome video! I’ve taken a handful of higher math classes (and watched a lot of math YT), but I’ve never seen number theory expressed as a semi module before! That makes so much of the random tidbits make sense, and is just so cool.

[isogash]

Factor space is great! It intuitively feels like there is so much potential mathematics to discover in relationships between numbers and patterns/functions in factor space. I haven't seen anyone else using it at all.

I started experimenting with it in relation to the Collatz conjecture and made a couple of interesting "discoveries." A really simple and fun one is that each factor has its own predictable fractal pattern as you contually apply the successor function. If you take the component of 2 of numbers 1, 2, 3, ... etc. you get the sequence 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ... etc. Every time you double the length of the sequence, you duplicate the existing sequence and then add one to the final element. This self-similarity in the sequence becomes very interesting when you start to think about integers with a potentially infinite number of digits (similar to p-adics) as they will still have a finite component of 2 (so long as they are a p-adic infinite power of 2.) It gives you a way to think about all possible numbers at the same time when attempting to prove theorems about repeated application of multiplication, such as with Collatz. When you keep dividing by 2, that component of 2 will hit zero, but your local space of integers on either side will share have a similar overall shape (hard to explain but if you picture the numbers as "peaks", the local neighbourhood will have the same peaks as your older neighbourhood did, just on a smaller scale with smaller numbers. In particular, the side on which the taller peaks are will not fundamentally change.)

Note that the sequence works in a similar way for higher factors, the sequence is just repeated self-similarly k times where k is the factor e.g. for 3 it's 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, ... etc...

Another interesting note on Collatz is you can actually change the (3n+1)/2 rule when n is odd into a new rule where only the multiplication is repeated:
1. add 1 to n
2. multiply n by 3 and divide by 2
3. repeated step 2 until n is odd
3. subtract 1 from n (and go back to dividing by 2 until n is odd again.)

After you add 1 to n, you can look at the component for 2 and it will tell you the number of times you are going to need to repeat step 2. This is because each repetition of step 2 is dividing by 2, so the component of 2 will fall until it hits zero and the number becomes odd. Interestingly, this means that the highest number of possible repeated (3n+1)/2 in a row is always the exponent of the next highest power of 2 greater than n, and the number with the most will always be the one that is 1 less than that power. Note that each repetition of step 2 is adding to the component of 3, which relatively moves the number within its range between powers of 2, unlike dividing by 2. (All combined, this means your local neighbourhood stays similar in the sense that the shape of the valley you are in will stay the same, but which side the taller peaks can be found on will change predictably.)

This all comes together to give you new ways to reason about the Collatz conjecture and what is really going on under the hood. It opens up the potential to define better understand which patterns are actually possible in the hailstone sequence and how patterns in the sequence in smaller numbers can relate to similar-looking patterns in larger numbers. In particular, you can see how there is ain interesting pattern to where numbers in lower intervals of powers of two will end up positioned on higher intervals.

Anyway, this is where it's really out of my depth, I don't have the mathematical skills or patience to take this any further and find exactly how the neighbourhoods change when the sequence is applied.

[eHobp]

Dude, I spent years thinking the factor space idea was an original thought of mine. Damn!!

[hgp]

love reading the comments and seeing how many other people independently thought to represent the naturals like this

[__christopher__]

The fact that gcd(a+b, a)=gcd(a+b, b)=gcd(a, b) means that for the sum of two arbitrary numbers, for those entries where they differ, the sum has the lower of the two as entry, while for those where they equal, the sum may have an equal or higher entry. The fact that a and a+1 are orthogonal is a special case of that.

[thelonglinest]

Like a lot of other commenters here, I independently thought of "factor space", and to me it's the most natural way to understand what multiplication actually is from an abstract algebraic point of view. I majored in math but I was kind of disappointed that it was never taught in my courses. Cool vid overall

[AnyVideo]

This is my favourite way to view the primes. I always want to see how many traditional properties of primes simply emerge from ordering these vectors without knowing anything about the underlying arithmetic.

Fundamentally, one could do the same with polynomials. A really nice case are F2 polynomials since they track neatly with binary numbers as well

[Inspirator_AG]

*@[**0:11**]:* They are all divisible by 30.

[WYXkk]

I think 'addition ruins factorization' explains better, which I think is true in all nontrivial cases. And number theory is just very about factorization.

[yb]

So highly informative ^_^ And the music that accompanied the introduction of the successor function is very fitting. Thanks for the explanations.

[iMíccoli]

I'm in the math Olympiad world and number theory is the hardest one for me because of how technical it is, you can't even understand the problems or even have an idea of what to do if you don't have experience nor any solid basics but when you get used to it's pretty manageable, you can solve problems and keep learning more without as much difficulty as as when you started. Also I love how mysterious it is sometimes but anyways good video ❤.

[alexwang]

There's a beautiful interpretation of the Lifting the Exponent lemma in this space

[MooImABunny]

since S is not linear on the vector space, I would avoid dropping the parentheses around the argument.
you write at a few points stuff like S²p, which is notation usually reserved for linear operations.
a quick proof that S isn't linear:
a²+1 ≠ (a+1)²
=> S(2A) ≠ 2S(A)
where A is the vector corresponding to a

[curtiswfranks]

Your factor space and its notation exactly matches my discovery/invention of the same thing in 2014. I am happy that I am not alone!

[shadeblackwolf]

3n+1. Whoever develops the theoreticap framework that solves this will be laying the groundwork for understanding addition and succession in this space.

[lumipakkanen]

Fun fact. Microtonal/xenharmonic music theorists affectionately call these vectors "monzos" and do all kinds linear and geometric algebra with them, slightly tweaking how large the prime numbers are. This results in tempering e.g. we can make it so that (3/2)^4 ~ 5/1 or equivalently [-4 4 -1> ~ [0 0 0> which is the Regular Temperament Theory way of specifying the meantone tuning system prevalent in Western music (meantone temperament predates the now ubiquitous 12-tone equal temperament which is itself a special case of meantone).