We will prove by contradiction. Assume that $2^n + 1$ is an odd prime but n is not a power of 2. Then, there exists an odd prime number p such that $p\mid n$ . Then, for some integer $k\geq 1$ ,

$n=p\times k.$

Therefore

$2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p.$

Here we will use the formula for the sum of odd powers, which states that, for $a,b\in \mathbb{R}$ and an odd positive number $n$ ,

$a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b+a^{n-3}b^2-...+b^{n-1})$

Applying this formula in 1) we obtain that

$2^n + 1=2^{p\times k} + 1=(2^{k})^p + 1^p=(2^k+1)(2^{k(p-1)}-2^{k(p-2)}+...-2^{k}+1)$ .

Then, as $2^k+1>1$ we have that $2^n+1$ is not a prime number, which is a contradiction.

In conclusion, if $2^n+1$ is an odd prime, then n must be a power of 2.

4 0

3 years ago

Find the value of x.

Over [174]

Answer:

no need to divide by 2.

Step-by-step explanation:

let's see u will understand.

solution,

x+67=180 (being straight line)

or, x=180-67

x=113 answer.

hope it'll help you.

7 0

3 years ago

Read 2 more answers

PLEASE HELP ME! I WILL GIVE BRAINLIEST

RSB [31]

Answer:

30 cm²

Step-by-step explanation:

The shaded area is the difference of the total area and the white triangle.

The white triangle has base of 9 cm and height of 4 cm.

Its area is:

A = 1/2*9*4 = 18 cm²

Area of rectangle:

A = 12*4 = 48 cm²

The shaded area:

48 - 18 = 30 cm²

3 0

2 years ago