MHB Is this a Gödel number of a Turing machine?

AI Thread Summary
The discussion revolves around verifying whether a specific large number can be classified as a Gödel number for a Turing machine based on its prime factorization. The number is expressed in terms of its prime factors, leading to a formulation of the Gödel number using specific parameters. A table is provided that lists pairs of indices alongside corresponding prime numbers, which are part of the factorization. The main inquiry is whether the presence of these primes in the factorization confirms the number as a Gödel number or if additional checks are necessary. There is also a note indicating that the approach to encoding Turing machines as numbers can vary, suggesting that the provided table may not align with standard Turing machine instructions. The discussion highlights the complexity of determining Gödel numbers and the need for clarity on the encoding methods used.
mathmari
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Hey! :o

We have the number \begin{align*}70&6737922567786324304462189150536772513339293263220644
\\ &=2^2\cdot 3\cdot 59^5\cdot 103\cdot 149^2\cdot 353\cdot 607\cdot 823^4\cdot 1409\cdot 1873^2\cdot 4201^3\end{align*}

I want to check if this is a Gödel number of a Turing machine.



From the prime factorization we have that $m=2$ and $k=1$.

The Gödel number is then of the form \begin{align*}G&=p_1^mp_2^k\prod_{i=1}^{(k+1)(m+1)}\prod_{j=3}^4p^{c_{ij}}_{\sigma_2(i,j)} \\ & = p_1^2p_2\prod_{i=1}^{2\cdot 3}\prod_{j=3}^4p^{c_{ij}}_{\sigma_2(i,j)} \\ & = p_1^2p_2\prod_{i=1}^6\prod_{j=3}^4p^{c_{ij}}_{\sigma_2(i,j)}\end{align*}

We have the following table:

$$(i,j) \ \ \ \sigma_2 \ \ \ P\sigma_2 \\
(1,3) \ \ \ 13 \ \ \ 41 \\
(1,4) \ \ \ 17 \ \ \ 59 \\
(2,3) \ \ \ 27 \ \ \ 103 \\
(2,4) \ \ \ 35 \ \ \ 149 \\
(3,3) \ \ \ 55 \ \ \ 257 \\
(3,4) \ \ \ 71 \ \ \ 353 \\
(4,3) \ \ \ 111 \ \ \ 607 \\
(4,4) \ \ \ 143 \ \ \ 823 \\
(5,3) \ \ \ 223 \ \ \ 1409 \\
(5,4) \ \ \ 287 \ \ \ 1873 \\
(6,3) \ \ \ 447 \ \ \ 3163 \\
(6,4) \ \ \ 575 \ \ \ 4201 $$ All the prime numbers of the factorization are in the table. Does this mean that the given number is the Gödel number of a Turing machine? Or do we have to check also something else? (Wondering)
 
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Your question makes little sense to someone unfamiliar with how Turing machines are coded as numbers in your course. You probably realize that there are dozens of ways to do it. Even the table does not look as standard Turing machine instructions.
 

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If this is a Gödel number, how can we determine the corresponding Turing table? (Wondering)
 
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