Halting problem in number theory

[Annatala Wolf]

Brief correction: the example is just a set of numbers. It is not the set
of proper divisors of 42 (for example, no 21).

On Tue, Dec 7, 2021 at 5:00 PM Annatala Wolf <a.lupine at gmail.com> wrote:

> Take the symmetric difference between the set of divisors of n and this
>> set where each element is multiplied by 2. Write the result in increasing
>> order. For any odd number in the list, type an opening parenthesis, for any
>> even number, type a closing parenthesis. The fact that this word is
>> well-matched was proved in [2].
>>
>> For example, take n = 12. The divisors of 12 are 1, 2, 4, 6, 12. Twice
>> these divisors: 2, 4, 8, 12, 24. The symmetric difference is 1, 6, 8, 12.
>> Hence, the parentheses are ( ( ) ).
>>
>
>
> The symmetric difference would be { 1, 6, 8, 24 }, but that doesn't change
> the fact that the parens are not generated based on divisibility by two as
> you state. That would make your paren string "()))" (in either case), which
> is not correct.
>
> Instead, the '(' paren appears for every divisor in the ascending
> symmetric difference and the ')' appears for every multiple of a divisor in
> that same sequence. Since 1 and 6 are divisors, while 8 and 24 are not, and
> 1 < 6 < 8 < 24, we have "(())".
>
> Here's an example from the book, using the real number e and the distinct
> proper divisors of 42:
>
> [image: image.png]
>
> No matter what you choose for your set of naturals and real under these
> parameters, there are three obvious guarantees: there exist the same total
> number of open as close parens, if there are any parens the first paren
> must be '(', and if there are any parens the last paren must be ')'. The
> nontrivial part of this requiring proof is showing that, for S = { proper
> divisors of any natural }, for any real number, and for every prefix of the
> resulting Dyck word, the number of ')' in the prefix does not exceed the
> number of '('.
>
> I haven't read enough into this yet to give a response to your specific
> query, but I wanted to clear that up first because I believe your code
> snippet is not mapping to the mathematics you're referring to.
>
> --
> /* Annatala Wolf, Lecturer
>  * Department of Computer Science and Engineering
>  * The Ohio State University
>  */
> enum E{A;System s;String t="/* Annatala Wolf, Lecturer%n * Department of
> Computer Science and Engineering%n * The Ohio State University%n */%nenum
> E{A;System s;String
> t=%c%s%1$c;{s.out.printf(t,34,t);s.exit(0);}}";{s.out.printf(t,34,t);s.exit(0);}}
>


-- 
/* Annatala Wolf, Lecturer
 * Department of Computer Science and Engineering
 * The Ohio State University
 */
enum E{A;System s;String t="/* Annatala Wolf, Lecturer%n * Department of
Computer Science and Engineering%n * The Ohio State University%n */%nenum
E{A;System s;String
t=%c%s%1$c;{s.out.printf(t,34,t);s.exit(0);}}";{s.out.printf(t,34,t);s.exit(0);}}
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