Prove that there are infinitely many primes of the form 4k + 3.
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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={p1,p2,…,pn}, where each pi 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(p1p2…pn) 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 pj be a prime factor of N that is of the form 4k+3. According to our initial assumption, pj must be one of the primes in the set P={p1,p2,…,pn}. This means pj divides the product p1p2…pn. If pj divides N and pj divides p1p2…pn, then pj must also divide their difference: N−4(p1p2…pn)=−1. This implies pj 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.