Quick one: \varphi{(p)}- \varphi{(p-1)}= [\varphi{(\varphi{(p)})}]^3

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In summary, the equation \varphi{(p)}- \varphi{(p-1)}= [\varphi{(\varphi{(p)})}]^3 represents the p-adic phi function in number theory, which calculates the difference between the Euler phi function of a prime number (p) and that of its predecessor (p-1), raised to the third power. The Euler phi function is defined as a number theory function that counts the number of positive integers less than or equal to a given number (p) that are relatively prime to it. The term [\varphi{(\varphi{(p)})}]^3 in the equation represents the third iterate of the Euler phi function, allowing for further manipulation and exploration of patterns
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find solutions where a) p is a prime; b) p is composite;

[tex]\varphi{(p)}- \varphi{(p-1)}= [\varphi{(\varphi{(p)})}]^3[/tex]

ps: it is not homework
 
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  • #2
How about 3? phi(3)=2, phi(2)=1.
 
  • #3
robert Ihnot said:
How about 3? phi(3)=2, phi(2)=1.

yes, is it the only one?
 

1. What does the equation \varphi{(p)}- \varphi{(p-1)}= [\varphi{(\varphi{(p)})}]^3 represent?

The equation represents a formula in number theory known as the "p-adic phi function". It calculates the difference between the Euler phi function of a prime number (p) and that of its predecessor (p-1), raised to the third power.

2. How is the Euler phi function (\varphi{(p)}) defined?

The Euler phi function is a number theory function that counts the number of positive integers less than or equal to a given number (p) that are relatively prime to it. In other words, it calculates the number of numbers between 1 and (p) that do not share any common factors with (p).

3. What is the significance of [\varphi{(\varphi{(p)})}]^3 in the equation?

The term represents the third iterate of the Euler phi function. This means that the phi function is applied to itself three times before being subtracted from the original phi function value. It is a way to further manipulate the values and explore patterns in the p-adic phi function.

4. Are there any known solutions to the equation \varphi{(p)}- \varphi{(p-1)}= [\varphi{(\varphi{(p)})}]^3?

Yes, there are known solutions to the equation. For example, when p = 2, the equation becomes \varphi{(2)}- \varphi{(1)}= [\varphi{(\varphi{(2)})}]^3, which simplifies to 1-1=0. This means that 2 is a solution to the equation. There are also other known solutions for different values of p.

5. What is the significance of the p-adic phi function in number theory?

The p-adic phi function is important in number theory as it can be used to study the distribution of prime numbers. It also has connections to other important number theory functions such as the Riemann zeta function and the Möbius function. The equation \varphi{(p)}- \varphi{(p-1)}= [\varphi{(\varphi{(p)})}]^3 is just one example of the many interesting properties and patterns that can be explored using the p-adic phi function.

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