To prove that there are no twin primes of the form $2^n - 1$ and $2^n + 1$ for $n > 2$ using modulo 3 arithmetic, let's proceed with the proof.

Proof:

Consider the numbers $2^n - 1$ and $2^n + 1$. We need to show that at least one of these numbers is not prime for $n > 2$ by examining their residues modulo 3.

Step 1: Examine $2^n \mod 3$

Since we are working with powers of 2 modulo 3, let's determine the pattern:

&2^1 \equiv 2 \pmod{3}, \\ &2^2 \equiv 4 \equiv 1 \pmod{3}, \\ &2^3 \equiv 2 \cdot 2^2 \equiv 2 \cdot 1 \equiv 2 \pmod{3}, \\ &2^4 \equiv 2 \cdot 2^3 \equiv 2 \cdot 2 \equiv 4 \equiv 1 \pmod{3}. \end{aligned}$$ From this, we see a repeating pattern: - $$2^{2k} \equiv 1 \pmod{3}$$ for any integer $$k$$, - $$2^{2k+1} \equiv 2 \pmod{3}$$ for any integer $$k$$. #### Step 2: Consider the forms $$2^n - 1$$ and $$2^n + 1$$ modulo 3 Let's consider the cases for $$n$$: - **Case 1:** $$n = 2k$$ (even $$n$$) If $$n$$ is even, then: $$2^n \equiv 1 \pmod{3} \implies 2^n - 1 \equiv 1 - 1 \equiv 0 \pmod{3},$$ $$2^n + 1 \equiv 1 + 1 \equiv 2 \pmod{3}.$$ So, $$2^n - 1 \equiv 0 \pmod{3}$$. This implies $$2^n - 1$$ is divisible by 3 and hence cannot be prime unless it is 3. For $$n > 2$$, $$2^n - 1$$ is greater than 3, so it is not a prime number. - **Case 2:** $$n = 2k + 1$$ (odd $$n$$) If $$n$$ is odd, then: $$2^n \equiv 2 \pmod{3} \implies 2^n - 1 \equiv 2 - 1 \equiv 1 \pmod{3},$$ $$2^n + 1 \equiv 2 + 1 \equiv 3 \equiv 0 \pmod{3}.$$ So, $$2^n + 1 \equiv 0 \pmod{3}$$. This implies $$2^n + 1$$ is divisible by 3 and hence cannot be prime unless it is 3. For $$n > 2$$, $$2^n + 1$$ is greater than 3, so it is not a prime number. #### Conclusion: In both cases, for $$n > 2$$, one of the numbers $$2^n - 1$$ or $$2^n + 1$$ is divisible by 3 and thus cannot be prime. Therefore, there are no twin primes of the form $$2^n - 1$$ and $$2^n + 1$$ for $$n > 2$$. Do you want details or have any questions? Here are some questions you might want to ask: 1. How do we prove other properties using modular arithmetic? 2. What are other examples of twin primes? 3. How does this proof extend to other forms of numbers? 4. Can you explain the pattern in powers of 2 modulo 3 in more detail? 5. What are some practical applications of modular arithmetic in cryptography? **Tip:** Understanding the behavior of numbers under different modulo operations can often simplify complex number theory problems.