Prime Numbers and Their Powers: A Comprehensive Analysis

Number theory, a fascinating branch of mathematics, delves into the properties and behavior of prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This article will explore a specific question related to prime numbers and their powers: if ( p ) and ( p^{21} ) are prime numbers, is ( p^{31} - 1 ) also a prime number?

Understanding the Problem

The question at hand is: if ( p ) and ( p^{21} ) are prime numbers, is ( p^{31} - 1 ) also a prime number?

Initial Observations

Let's first consider the expression ( p^{31} - 1 ). We can rewrite this expression as follows:

[ p^{31} - 1 (p - 1)(p^{30} p^{29} cdots p 1) ]

Here, the second factor is a sum of 31 terms, each of which is a multiple of ( p ) plus 1. This form immediately suggests that ( p^{31} - 1 ) is not a prime number, as it has two factors, ( p - 1 ) and the sum of the terms.

General Case Analysis

Let's generalize the observation. Consider the expression ( p^{31} - 1 ) for any natural number ( p > 1 ). The expression can be factored as:

[ p^{31} - 1 (p - 1)(p^{30} p^{29} cdots p 1) ]

This factorization is always valid, and unless ( p 2 ) and ( p^{21} ) is prime, the factor ( p - 1 ) will be greater than 1, making ( p^{31} - 1 ) a composite number.

Example and Verification

Let's verify this with a concrete example. Consider ( p 2 ):

[ 2^{21} 2,147,483,648 ] (which is not a prime number)

We need to check if ( 2^{31} - 1 ) is a prime number:

[ 2^{31} - 1 2,147,483,647 ] (which is not a prime number)

This can be verified by noting that:

[ 2^{31} - 1 (2 - 1)(2^{30} 2^{29} cdots 2 1) 1 times (2^{30} 2^{29} cdots 2 1) ]

The second factor is clearly greater than 1 and is a sum of 31 terms, each of which is 1 more than a power of 2. Therefore, ( 2^{31} - 1 ) is a composite number.

Conclusion

In conclusion, if ( p ) and ( p^{21} ) are prime numbers, ( p^{31} - 1 ) is always a composite number. This is because ( p^{31} - 1 ) can be factored into ( (p - 1)(p^{30} p^{29} cdots p 1) ), and both factors are greater than 1 as long as ( p > 2 ).

The same reasoning applies to any natural number ( p > 1 ), as ( p^{31} - 1 ) can always be written in the form of a product of two factors greater than 1.