- Diophantine equations are as they say Turing complete. That is for any question about does this Turing machine with this tape halt with a certain value there is a corresponding Diophantine equation, which has solutions if the machine halts with the output corresponding to the values it is solved by. I think this paper covers it for register machines rather than Turing machines directly: https://carleton.ca/math/wp-content/uploads/Nick-Murphy-Hono...
- The article doesn't really tell us much about the "why" unfortunately. Diophantine equations are introduced but all the interesting stuff is promised in future articles which haven't come yet. All the reader can take from this is that these equations lead to some "profound hidden structures" without a good idea what they are.
I get that it's hard to wrap one's head around the Langlands program but I'd love to see at least more exposition on the following statement:
>inventing the Euclidean algorithm is essentially equivalent to inventing unique prime factorization
- One context in which diophantine equations arise is hidden within the innards of loop optimizing compilers, where loop carried dependencies are considered, as they constrain parallelization.
I had (and donated to an engineering library in Urbana) a book about just this from the early 90s. I tried finding it on Amazon but no such luck.
This was a recurrent tool at
https://en.wikipedia.org/wiki/University_of_Illinois_Center_...
- The purpose of this article was, secretly, to tell the reader about another class of Diophantine equations which leads to the Langlands program, which studies from incredibly intricate hidden structure inside of number theory. The Langlands program studies Diophantine equations of the form
This is not what the Langlands program is
