Halting problem in number theory

[Annatala Wolf]

>
> 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);}}
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