Prove that there are infinitely many primes of the form 4n3

Prove that there are infinitely many primes of the form 4n+3; of the form 6n+5.

Solution

Answer :

( 1 ) first we show that \" there are infinitely many primes of the form 4n+3 \"

      we use proof by contradiction

     Assume we have a set P of finitely many primes of the form 4n+3

      where P = { p1,p2,...,pn }.

     now we Construct a number N such that

    N = 4 * p1*p2*...*pn – 1

= 4 [ (p1*p2*...*pn) – 1 ] + 3

then N can either be prime or composite.

If N is a prime, there’s a contradiction since N is in the form of 4n + 3 but does not equal to any of the number in the set P = { p1,p2,...,pn }.

If N is a composite, there must exist a prime factor “a” of N such that a is in the form of 4n+3.

All the primes are either in the form of 4n+1 or in the form of 4n+3. If all the prime factors are in the form of 4n+1,

N should also be in the form of 4n +1. There should exist at least one prime factor of N in the form of 4n+3.

“a” does not belong to set P.

Then N/a = (4 * p1*p2*...*pn– 1) / a

      = (4 * p1*p2*...*pn) / a - 1/a

(1/a is not an integer)

Conclusion:

a is a prime in the form of 4n+3, but a does not belong to set P.

Therefore, we proved by contradiction that there exists infinitely many primes of the form 4n+3.

( 2 )

Now we show that \" there are infinitely many primes of the form 6n + 5 \"

      we use proof by contradiction

     Assume we have a set P of finitely many primes of the form 6n + 5

      where P = { p1,p2,...,pn }.

     now we Construct a number N such that

    N = 6 * p1*p2*...*pn – 1

= 6 [ (p1*p2*...*pn) – 1 ] + 5

then N can either be prime or composite.

If N is a prime, there’s a contradiction since N is in the form of 6n + 5 but does not equal to any of the number in the set P = { p1,p2,...,pn }.

If N is a composite, there must exist a prime factor “a” of N such that a is in the form of 6n + 5.

All the primes are either in the form of 6n + 1 or in the form of 6n + 5.

If all the prime factors are in the form of 6n + 5, N should also be in the form of 6n + 5. There should exist at least one prime factor of N in the form of 6n+5.

“a” does not belong to set P.

then N/a = (6 * p1*p2*...*pn– 1) / a

      = (6 * p1*p2*...*pn) / a - 1/a

(1/a is not an integer)

Conclusion:

a is a prime in the form of 6n + 5, but a does not belong to set P.

Therefore, we proved by contradiction that there exists infinitely many primes of the form 6n + 5.