Is p^n Deficient if p is a Prime?

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Homework Help Overview

The discussion revolves around the concept of deficient numbers in number theory, specifically examining whether \( p^n \) is deficient when \( p \) is a prime number.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of deficient numbers and consider the divisors of \( p^n \). Questions arise regarding the sum of these divisors and how they relate to the definition of deficiency.

Discussion Status

There is an ongoing exploration of the properties of prime numbers and their divisors. Some participants suggest starting with definitions and applying them to the problem, while others reiterate the relationship between the divisors of \( p^n \) and the criteria for deficiency.

Contextual Notes

Participants are discussing the implications of \( p \) being prime and the specific nature of its divisors in the context of the problem. There is a focus on understanding the definitions involved without reaching a definitive conclusion.

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Homework Statement



Show that if p is a prime, then p^n is deficient.

Homework Equations





The Attempt at a Solution



I have no idea where to start.
 
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You could start with the definition of a deficient number.
If p is prime, it's only divisors are 1 and p and thus, the sum of it's divisors is p+1.
What are the divisors of pn?
 
VeeEight said:
You could start with the definition of a deficient number.
If p is prime, it's only divisors are 1 and p and thus, the sum of it's divisors is p+1.
What are the divisors of pn?

would be P^n +P^1+P^0 correct?
 
If p is prime, then it's only divisors are 1 and p. The divisors of pn are also 1 and p, which sum to p+1. Apply this to the criteria of a deficient number.
 

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