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Show that there are infinitely many primes of th form 4n+3

Show that there are infinitely many primes of th form 4n+3

Grade:9

2 Answers

Arun
25757 Points
5 years ago
Dear student
 
We emulate Euclid’s argument for the existence of infinitely many primes:
Let’s argue by contradiction. Suppose there are finitely many primes of the form
4n + 3 and they are exactly {p1, . . . , pk}. Consider N = (p1 · · · pk)
2 + 2. Then
N ≡ 3 (mod 4). On the other hand N is odd and is not divisible by any pi
. It
follows that all prime divisors of N are congruent to 1 modulo 4. Conclude N ≡ 1
(mod 4). A contradiction!
 
Regards
Arun (askIITians forum expert)
Atharv Sagar Suryawanshi
25 Points
5 years ago
 Sir I didn't understood the last argument N is not divisible by pi . Is i>k?
As N is product of primes then it should be divisible by pi right?

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