Are those huge numbers that Roger Penrose wrote in his book the smallest possible numbers for the Turing Machine Number of the Universal Turing Machine?

No, they are not.

You are obviously referring to the 5,495-digit binary number on pages 93 through 96, or the 1,654-digit denary number on page 74 of the book The Emperor's New Mind by Sir Roger Penrose.

These are not the smallest possible Turing Machine Numbers for the Universal Turing Machine, and I will now show you how you can generate smaller Turing Machine Numbers for the same Universal Turing Machine that Sir Roger Penrose had in mind. Just follow these steps:

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Step 1: Download the Notation Converter from the Home page of this website and open the Excel file you just downloaded.

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Step 2: Click on the picture.

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Step 3: When asked From which numerical notation will you be converting your number? select Denary.

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Step 4: When asked What Denary number shall we use? select Example from the book "The Emperor's New Mind".

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Step 5: When asked Which Denary Number Example from "The Emperor's New Mind" would you like to use? select From page 74, the Denary Number of a Universal Turing Machine.

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Step 6: When you see So Far So Good! click OK.

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Step 7: When you see Well Done! click OK.

Thus far, you have successfully converted the Denary Number of the Universal Turing Machine into its Machine States. These are in rows 16 and 17 of the Excel file.

What we would like to do now is to convert these two rows of Machine States into a Denary Number and see what we get. (Try to remember the name of the sheet on which you generated these Machine States. If you had started with a new Notation Converter that you just downloaded from the Home page, then its name should be 1.)

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Step 8: Go to the Welcome sheet. You do that by clicking on the Welcome tab in the lower left of your screen.

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Step 9: Click on the picture.

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Step 10: When asked From which numerical notation will you be converting your number? select Machine State.

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Step 11: When asked What Machine States shall we use? select I will provide my own Machine States.

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Step 12: When asked How will you provide your Machine States? select From a sheet in this Excel Workbook.

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Step 13: When you see a pop-up window, read the directions, and then select the sheet with the Machine States you generated in steps 6 and 7. (The name of that sheet is probably 1.) Select that sheet by clicking on its tab along the bottom of the screen. Then using your mouse, highlight rows 16 and 17. Then, click OK in the pop-up window.

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Step 14: When you see So Far So Good! click OK.

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Step 15: When you see Well Done! click OK.

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Notice that you now have the Denary Number of the Universal Turing Machine in row 17 of your Excel file. Notice also that it is different from the Denary number that you started with. In fact, this number has only 1,650 digits, compared with the 1,654 digits that were on page 74 of the book The Emperor's New Mind.

(A difference of 4 digits doesn't seem like much, until you think of it as being less than 0.0016% of the original number!)

Of course, that same number in Binary Notation is smaller as well, having only 5,479 digits, compared with the 5,495 digits found on pages 93 through 96 of the same book.