Digital root of prime numbers

By Kardi Teknomo, PhD.

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In this section, I describe my exploration on the pattern of digital root of prime number.

Prime numbers are integers that have only factors: 1 and the number itself. The list of the first few primes : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, …

All primes are odd numbers except 2. Number 2 is the only even prime.

How about digital root of primes? Can we find a pattern from the digital root of primes?The list of the first few primes with its digital root:

Then the pattern of the first 15 primes is 235724815241572

The list of the first 664,579 primes number (that is primes below 10 million) with its digital root and additive persistence can be downloaded as a text file here (Warning: file size 32.3 MB).

Based on that limited number of primes and its digital root, I would say that the digital root of prime numbers does not form any cyclical or repeated pattern. You can download the pattern of digital root of the first 664,579 primes here (1.3 MB).

Here are my conjectures regarding digital root of prime numbers:

1. The pattern of digital root of prime number is chaotic (I am not sure how to prove this?)
2. Number 3 (which of course the digital root is also 3) is the only prime that has digital root 3 (surprise!)
3. There no prime has digital root of 6 and 9.
4. Digital root of prime numbers are almost equally distributed among (1, 2, 4, 5, 7, 8), about 1/6 (limiting behavior in the long run)

Of course, my conjectures above are based only on a few primes (only 664,579 primes) and there is no guarantee that conjectures above are true for a very large prime.

Here is the distribution of the digital root of prime number

After analyzing the pattern of digital root of prime number, I have more questions than answers. Some of the few crucial questions would be:

• Is there any way to find cyclical pattern of a large number?
• Since prime number seems to be infinite, how to proof that my conjectures above are correct or false?
• Is there a cleverer way to find digital root of very large prime number?
• Why there is only one prime has digital root of 3
• Why there is no prime has digital root of 6 and 9?

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These tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2005). Digital Root. http:\\people.revoledu.com\kardi\tutorial\DigitSum\