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sean_pedersen
Who can find the mistakes?
Someone
The text is not 100% clear to me (partly because it is a bogus proof), but you seem to redefine the arithmetic operators as

  i ‘+’ j := i + √j
  i ‘-‘ j := i - √j
  i ‘×’ j := i + j
  i ‘÷’ j := i - j
That completely changes the structure of the ‘integers’. Essential properties such as

  i + j = j + i
  (i + j) + k = i + (j + k)
  i + i = 2 × i
no longer hold, so you can’t use the usual rules of arithmetic. For example, the PDF does

  ‘3’ × X ‘+’ ‘4’ = (‘3’ × X) ‘+’ ‘4’ = (‘3’ + X) + ‘4’ = (‘3’ + X) + ‘√4’
and then uses the rule

(‘3’ + X) + ‘√4’ = (X + ‘3’) + ‘√4’

but that rule doesn’t hold in the alternative arithmetic.

Also, ‘even’ and ‘odd’ are completely different from the traditional even and odd.

Fo example, for any even ‘n’, we have

  ‘n’ ÷ ‘2’ = n - 2
so division by two of an ‘even’ ‘number’ always keeps the number ‘even’, and for starting point ‘4’, we get an infinite sequence ‘2’, ‘0’, ‘-2’, etc. (also a direct counterexample to the claim that you always end up at ‘1’)
sean_pedersen
Ah ok thanks for the corrections, so I need to evaluate how my 2D (projected) arithmentic works for the proof to work out. Thank you so much for taking my work seriously.
sean_pedersen
You only criticized my first proof though. I shipped two proofs gladly! The second one just uses inverses of the collatz function and graph theory.
Someone
The second one is inscrutable to me, but seems to depend on the existence of the inverse of a single iteration of the Collatz sequence. That inverse doesn’t exist. An easy counterexample is that both 1 and 8 in a single step produce 4.

In general, for any odd n, both n and 6n+2 in a single step lead to 3n+1.

sean_pedersen
"That inverse doesn’t exist." That is my new math contribution apparently then. I am happy to explain my proof.

I can inverse f(x)=f(x+1) easily as g(x)=f(x)-1 since g(f(x)) = x do you understand my functional inversion approach now better?

Khelavaster
What's the reverse of '4', since '1' and '8' both link to it?
sean_pedersen
The reverse is a "Quantum" tree (DAG): 4->1,4->8
sean_pedersen
I need an edorsment on arXiv to post it there in the math section... Can anyone help me? Here is my arXiv endorsment code: 9C8E9J