User blog comment:Eners49/The secret 0th hyper-operator?/@comment-2601:142:2:EC49:A496:CA1D:3DA2:B5DD-20180728144329

Here's some more x?y facts I came up with:

WARNING: THIS IS A BIT OF AN ESSAY.

It's easy to prove that x(y^s)x = x(y+1^s)2, which means that x?x = x+2. This means that 4?4 equals 6.

This creates a bit of a conundrum however, as 4+4 can be shown to be equal to 4?(4?(4?4)), or 4?(4?6). This is weird, as that means that 4?4 is as far away from 4 as 4?(4?(4?4)) is from 4?4, which defies common logic. To get any kind of meaning out of this, we need to figure out how big 4?6 is so we can derive a pattern.

That seems impossible, however, 4?6 can be written as 4?(4?4) which can be written as 4+3, which means huzzah! We have proven that 4?6 equals 4+3 equals 7! This also means that 4?7 equals 8, using the same reasoning.

In fact, this knowledge that chained ? notation equals addition, and that x?x = x+2 means that we can prove that we can endlessly prove these numbers, which means we can come up with a formula! We can show that x?(x+y) = x+(y+1) if y is at least 2.

Now that we have a (decent) grip on x?y, we can start to apply these same principles to the -1th hyper-operator, which I shall represent with x??y.

Lets list all the ? numbers we have integer values for in ?? form to try to see how x??y works.

2??2??2??2 = 5,

2??2 = 4

3??3??3 = 5

3??3??3??3??3 = 6

4??4??4??4 = 6

4??4??4??4??4??4 = 7

4??4??4??4??4??4??4 = 8

That doesn't answer any questions anybody had ever. I'm stumped.

Ok, so let's talk about the number 1 for a second. and then come back to the ?? function.

Those equations don't help us because they have several inputs, and we're trying to look for equations that have 2.

This is hard, as the formulas/shortcuts for x?y only work if y is equal to or larger than x, which means having x's that are at least 2 doesn't help us, as the y needs to be equal to 2 for us to figure anything out.

Well, what about one then? One's smaller than 2! Well if we use the x?x rule on one, it can be shown that 1?1 equals 3, which when put into ?? notation means that we have shown that: 1 = 3.

1 = 3.

Yeah that is not how anything works ever.

In conclusion, non-positive hyper-operators break the rules of everything, and therefore are nonsense.

Feel free to use my discoveries to make your own discoveries, just know that you're entering a world of "dark math" where things make sense only if you ignore the things that don't.