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What is the significance of Mochizuki's ABC Conjecture?

As someone who does not have a backing in hard mathematics, is there a wildly simplified version of what the abc conjecture is/and does for mathematics?
gpt
It helps us understand a little bit about what happens to primes under addition. 

If you have two numbers that you know the prime decomposition of, then you know the prime decomposition of their produce automatically. 5^(3)\*3\*2^(2) multiplied by 7^(2)\*5 is 7^(2)\*5^(4)\*3\*2^(2). Primes behave very well and predictably under multiplication. 

Primes do not behave well under addition. If you know the decomposition of two numbers, then you know nothing about the prime decomposition of their sum. 13 and 3 are both prime, but 13+3=16=2^(4), is a relatively huge power of some unrelated prime, there was no way we could predict that just from 13 and 3. It seems like it is a lost cause trying to mix the ideas of primes with addition.

The ABC-Conjecture tells us otherwise. It tells us how the situation is biased. It can tell us that if A+B=C, and A and B have lots of bit exponents in their prime decomposition, then C will likely *not* have big exponents in its prime decomposition. It says much more than this, and it's statement is very precise, but this illustrates how it gives constraints and limitations on how primes can behave under addition, which is something we've never had before.

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This is used in Number Theory, because most of Number Theory can be boiled down to finding integer solutions to algebraic equations. For instance, can we find integer solutions to 5x+7y=1? A fundamental theorem says that we *can* (eg x=10,y=-7). The integer solutions to x^(2)+y^(2)=z^(2) are Pythagorean Triples (eg, x=3,y=4,z=5), and these have inspired a lot of math. The question of whether or not integer solutions exist to x^(n)+y^(n)=z^(n), for n>2, is Fermat's Last Theorem and it basically shaped the course of Number Theory for 350 years. Integer solutions on conics (ax^(2)+bxy+cy^(2)=d), elliptic curves (y^(2)=x^(3)+ax+b) and so many more, equations like these are the foundation to Number Theory.

Note how they all have a "+" in there. Because of this, there is a lot of machinery about primes, that we take for granted when it comes to just multiplication, that is unavailable to us. It is hard to solve these equation, hard to determine whether or not there can actually be solutions and even harder to figure out exactly how many solutions there are to them. Many deep theorems are partial answers to some of these questions. The ABC-Conjecture gives us a way to make these questions easier. If we have this way to work with primes and addition, it becomes easier to talk about what the solutions can look like. For instance, using it we can reduce the original monster proof of Fermat's Last Theorem to just a couple of lines.

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Finally, the ABC-Conjecture is just a piece of a much larger framework in the field of "Arithmetic Geometry" which, roughly, tries to create a framework where geometry and arithmetic become indistinguishable and where you can use methods of one to solve problems in the other. It's a pretty deep field and has a lot of active research in it, but it's really ambitious and we're mostly guided by hints, feelings and analogies between arithmetic and geometry that we really want to be able to write down as solid theorems using solid definitions. This is the context in which Mochizuki really works. He takes ideas formulated by people throughout the 1900s about what arithmetic does, what geometry does and how they kinda interact, and turns the volume up to 11. So not only does Mochizuki's work promise a proof for the ABC-Conjecture, but it is quite possibly an arena for Arithmetic Geometry to really flourish and take footing in.