Okay, so I have a lot of anxiety about my ability to do high level mathematics. Like, I'm applying right now for my masters but I'm concerned about my ability to keep up (which is why I'm doing the masters track not the PhD track in case I need to dip).
BUT THEN. I see papers like this one and it gives me hope. The paper attempts to prove that Euler's gamma constant is transcendental. This means it is NOT an algebraic number which are numbers that are solutions to polynomial equations with integer coefficients. It's not necessary to understand that but I'll come back to that.
So they "prove" in the paper that if I have a sum of real numbers and one is transcendental then the result must be transcendental. AND THEY PROVE IT. LIKE THIS.
Let B be transcendental and A be a number (transcendental or not). A+B is obviously transcendental because B is transcendental therefore A+B is transcendental. Yes, the thing I want to prove is true because the thing I want to prove is true. The verbatim text with annotation in brackets:
A+B cannot be written as a solution for polynomial equations with integer coefficients (cannot be algebraic--eg is transcendental), since
B is transcendental, and therefore A+B is transcendental.
B is transcendental, and therefore A+B is transcendental.
Watch as I elegantly disprove this statement with 3 lines:
1. Let k=1-pi, whether k is transcendental is irrelevant.
2. pi is transcendental
3. pi+k=1 which is not transcendetal.
The sum of a transcendental and another number is not necessarily transcendental. I'd be willing to accept the possibility that a transcendental plus an algebraic are transcendental but that's not what's being "proven".
There are so many other errors in this proof, but I just can't grasp how the hell someone thought to put that up online without even checking if what they were proving had a counter example. My first thought reading that was "... hmmm. Idk if I'm sold, let me check". How could you write that paper and go "yup looks good, no need to look any deeper".
Like, math is hard, I get it, but this takes an absolutely busted level of confidence. I'm certainly impressed for that, but the mathematical prowess is something to be desired.
