Let p_j be a prime factor of N that is of the form 4k+3. According to our initial assumption, p_j must be one of the primes in the set P = p_1, p_2,, p_n. This means p_j divides the product p_1 p_2 p_n. If p_j divides N and p_j divides p_1 p_2 p_n, then p_j must also divide their difference: N - 4(p_1 p_2 p_n) = -1. This implies p_j divides -1, which is impossible for a prime number.

Let p_j be a prime factor of N that is of the form 4k+3. According to our initial assumption, p_j must be one of the primes in the set P = p_1, p_2,, p_n. This means p_j divides the product p_1 p_2 p_n. If p_j divides N and p_j divides p_1 p_2 p_n, then p_j must also divide their difference: N - 4(p_1 p_2 p_n) = -1. This implies p_j divides -1, which is impossible for a prime number.

Prove that there are infinitely many primes of the form $4k + 3$ .

Answer: There are infinitely many primes of the form $4k + 3$ .

Step-by-step solution

Step 1: Assume a finite number of primes

To prove that there are infinitely many primes of the form $4k+3$ , we use proof by contradiction. We start by assuming that there is a finite number of such primes. Let this finite set of primes be $P = \{p_1, p_2, \dots, p_n\}$ , where each $p_i$ is of the form $4k+3$ .

Step 2: Construct a new number N

Now, let's construct a new number $N$ . We define $N$ as four times the product of all these primes, minus one. This number $N$ can also be written in the form $4k+3$ , since $4(p_1 p_2 \dots p_n)$ is a multiple of 4, and subtracting 1 gives a number of the form $4k+3$ .

Step 3: Analyze the prime factors of N

Every prime factor of $N$ must be of the form $4k+1$ or $4k+3$ . If all prime factors of $N$ were of the form $4k+1$ , then their product $N$ would also be of the form $4k+1$ . However, $N$ is of the form $4k+3$ . Therefore, $N$ must have at least one prime factor of the form $4k+3$ .

Step 4: Contradiction

Let $p_j$ be a prime factor of $N$ that is of the form $4k+3$ . According to our initial assumption, $p_j$ must be one of the primes in the set $P = \{p_1, p_2, \dots, p_n\}$ . This means $p_j$ divides the product $p_1 p_2 \dots p_n$ . If $p_j$ divides $N$ and $p_j$ divides $p_1 p_2 \dots p_n$ , then $p_j$ must also divide their difference: $N - 4(p_1 p_2 \dots p_n) = -1$ . This implies $p_j$ divides $-1$ , which is impossible for a prime number.

Step 5: Conclusion

Since our assumption leads to a contradiction, the initial assumption that there is a finite number of primes of the form $4k+3$ must be false. Therefore, there are infinitely many primes of the form $4k+3$ .

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