# Interesting procedure

I have played with several numbers with the following rules:

• Take a number. (Example 10)

• Find the prime factors of the number (if it have) . ( Example 10 have prime factors 2 and 5)

• Make a number using those prime factors in ascending order. (Example for 10 it will be 25 as prime factors are 2 and 5)

• Now continue the process with the new number you have got. Continue this process until you get a prime number as it will not have any prime factors

Applying the whole process on 10 :

10 => 25 => 55 => 511 => 773

I have applied this procedure on many numbers. And I concluded a statement:

All numbers will eventually fall to a prime through the process

For 8 it was a quiet big task:

8 => 222 => 2337 => 31941 => 33371313 => 311123771 => 7149317941 => 22931219729 => 112084656339 => 3347911118189 => 11613496501723 => 97130517917327 => 531832651281459 => 3331113965338635107

Another incredible number was 20 and I am unable to obtain a prime yet. You can try it yourself.

You should also try out if you can find any number not falling to a prime, or if you can prove or disprove my statement.

Note by Sahar Bano
1 year, 2 months ago

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This is quite intriguing Sahar! I immediately decided to try writing a program which would go through this process, but quickly realized that finding the prime factorization of numbers larger than $100,000,000$ is quite taxing on a laptop! It took mine about a minute just to get to the fifth iteration for 8!

However, here is something I observed. When I set a cutoff in my program which ended the process when it hit numbers larger than $20,000,000$, here are the numbers which were ended between 2 and 200:

8, 16, 20, 40, 44, 48, 49, 64, 65, 77, 78, 80, 81, 86, 87, 91, 96, 104, 105, 112, 116, 120, 123, 124, 126, 128, 129, 130, 132, 135, 136, 144, 146, 150, 160, 161, 164, 168, 169, 170, 176, 178, 180, 184, 185, 186, 188, 190, 192, 195, 196

Interestingly, many of these numbers come in consecutive pairs (for instance 77 and 78). This could easily change for higher cutoff values of course.

Hopefully someone else has some more insight!

- 1 year, 2 months ago