Prime Numbers: (2^n - 1) and (2^n + 1)

[smithg86]

1. The problem statement, all variables and given/known data

I was able to prove both of these statements after getting some help from another website, but I am trying to find another way to prove them. Can you guys check my work and help me find another way to prove these, if possible? Thanks.

Part A: Show that if 2^n - 1 is prime, then n must be prime.

Part B: Show that if 2^n + 1 is prime, where n \geq 1, then n must be of the form 2^k for some positive integer k.

2. Relevant equations

(x^k) - 1 = (x - 1)*(x^(k-1) + x^(k-2) + ... + x + 1)

3. The attempt at a solution

Part A:

Write the contrapositive,
n is not prime (a.k.a. n is composite) ==> 2^n - 1 is composite
Assume n is composite. Let n = p*q, where neither p nor q are 1.
Then,
2^n - 1 = (2^p)^q - 1 = (2^p - 1)*((2^p)^(q-1) + (2^p)^(q-2) + ... + (2^p) + 1)

Note that 2^p - 1 > 1. Also, ((2^p)^(q-1) + (2^p)^(q-2) + ... + (2^p) + 1) > 1. So we have factored 2^n - 1, thus it is not prime. We have proved the contrapositive, so the original statement is true.

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Part B:

Note that if n is of the form 2^k, then n's prime factorization is only composed of 2's. Thus, the contrapositive of the original statement is as follows:

n = b*(2^k), where b is a positive odd number ==> 2^n + 1 is composite.

Let n = b*(2^k). Then,

2^n + 1 = 2^(b*(2^k)) + 1 = ((2^(2^k))^b + 1 = (2^(2^k) + 1)*{[(2^(2^k)]^(k-1) + [2^(2^k)]^(k-2) + ... + [2^(2^k)] + 1}

Observe that [2^(2^k) + 1)] > 1 and {[(2^(2^k)]^(k-1) + [2^(2^k)]^(k-2) + ... + [2^(2^k)] + 1} > 1. We have factored 2^n + 1, so it is composite. This proves the contrapositive of the original statement, so the original statement is true.

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Is there another way to prove either one of these statements?

[Kummer]

Is there another way to prove either one of these statements?

Why would you want to? This is both a short and elementary solution. That is often considered to be the nicest proof.

[smithg86]

Why would you want to? This is both a short and elementary solution. That is often considered to be the nicest proof.

I thought it would provide more insight as to why the statements are true.

[americanforest]

[b]
Part B:

Note that if n is of the form 2^k, then n's prime factorization is only composed of 2's. Thus, the contrapositive of the original statement is as follows:

n = b*(2^k), where b is a positive odd number ==> 2^n + 1 is composite.

Let n = b*(2^k). Then,

2^n + 1 = 2^(b*(2^k)) + 1 = ((2^(2^k))^b + 1 = (2^(2^k) + 1)*{[(2^(2^k)]^(k-1) + [2^(2^k)]^(k-2) + ... + [2^(2^k)] + 1}

Observe that [2^(2^k) + 1)] > 1 and {[(2^(2^k)]^(k-1) + [2^(2^k)]^(k-2) + ... + [2^(2^k)] + 1} > 1. We have factored 2^n + 1, so it is composite. This proves the contrapositive of the original statement, so the original statement is true.


Should it state

[(2^{2^{k}})^{b}+1]=(2^{2^{k}} + 1)([2^{2^{k}}]^{b-1} + [2^{2^{k}}]^{b-2} + ... + [2^{2^{k}}] + 1)?

Can someone prove the general case of this expansion via Binomial Theorem for me?

[General_Sax]

(x^k) - 1 = (x - 1)*(x^(k-1) + x^(k-2) + ... + x + 1)


Where does this factorization come from? I just need a link or something. Thanks.