How to compare numbers in primoradic notation?

Question

What is the simplest solution to comparing which of two numbers is greater than or less than the other when they are both in primoradic notation.

The easiest way is to remove the common factors. But how do we compare what's left?

When is 1 unit of the xth prime greater than 2 of the "x minus y" prime hold true? Is there some universal properties that we can use to help aid this process?

Are there cases above a certain xth prime where we get properties that become helpful? What about before a certain xth prime?

Answer 1

You can compare them in a way similar to how you'd naturally compare two decimal numbers: the number with the most digits is the greater of the two; if they have the same number of digits, then working left-to-right(→) the one with the larger digit will be the greater.

Working with the digits right-to-left(←):

• the numerical base for the digit is the primorial of the position
• the digit itself is restricted to a natural number between zero and one less than the next prime in the primorial of the next higher position to the left