Question

prove that there are infinitely many primes of the form 4k+3

Answer 1

prove that there are infinitely many primes of the form 4k+3

• Proof By Contradiction
• Assumption: Assume we have a set of finitely many primes of the  form 4k+3
• P = {p1, p2, …,pk}.
• Construct a number N such that
• N = 4 * p1* p2* … *pk – 1
• = 4 [ (p1* p2* … *pk) – 1 ] + 3
• N can either be prime or composite.
• If N is prime, there’s a contradiction since N is in the form of 4k +3 but does not equal to any of the number in the set P.
• If N is a composite, there must exist a prime factor “a” of N such  that a is in the form of 4k+3.
• All the primes are either in the form of 4k+1 or in  the form of 4k+3. If all the prime factors are in the  form of 4k+1, N should also be in the form of 4k +1. There should exist at least one prime factor of  N in the form of 4k+3.
• “a” does not belong to set P
• N/a = (4 * p1* p2* … *pk – 1) / a
• = (4 * p1* p2* … *pk ) / a - 1/a (1/a is not  an integer)
• Conclusion:
• a is a prime in the form of 4k+3, but a does not belong to set P.
• Therefore, we proved by contradiction that there exists infinitely  many primes of the form 4k+3.

Answer 2

Answer:

glzodivfl cup dah KLMझ चेऑजृडन p gh i gf up