Prime division & repetition period

In summary, the conversation discusses the concept of prime numbers and a function that determines the period of repetition in the decimal expansion of a rational number. It is shown that this period is independent of the specific prime number used and a separate function is defined to represent this. It is then observed that when the result of this function is used in a certain equation, it always results in a unit fraction. The reason for this is because the number system used is base 10.
  • #1
caveman1917
33
3
Something odd i noticed while playing around with primes.

We have the set of prime numbers P and a p ∈ P.
Define a function f:Q → N that will give the period of the repetition in the decimal expansion of some number r ∈ Q.

1) ∀ p ∈ P: ∃ n ∈ N: ∀ q ∈ P, q < p: f(q/p) = n.
So n is independant of q.

So define a function g:N → N: ∀ p ∈ P, ∀ q ∈ P, q < p: g(p) = f(q/p).

2) ∀ p ∈ P: ((p - 1) / g(p)) ∈ N.

You'll always get a unit fraction 1/2, 1/3, 1/4... never something like 5/7.
I was wondering why?
 
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  • #2
Because the number system we use is base 10 :eek:)
 

What is prime division?

Prime division is the process of breaking down a number into its smallest prime factors. This is done by finding all the prime numbers that can divide evenly into the original number.

What is the repetition period of a prime number?

The repetition period of a prime number is the number of digits in the repeating pattern when the number is expressed as a decimal. For example, the repetition period of 1/3 is 1, as the decimal form is 0.3333...

Why is prime division important?

Prime division is important in various mathematical and scientific fields, such as cryptography and number theory. It also has practical applications in simplifying fractions and finding common factors between numbers.

Can any number be broken down into its prime factors?

Yes, every positive integer can be expressed as a unique product of prime numbers. This is known as the fundamental theorem of arithmetic.

How is prime division related to prime factorization?

Prime division is the process of finding the prime factors of a number, while prime factorization is the expression of a number as a product of its prime factors. Prime factorization is essentially the result of prime division.

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