Problem 1 Two prime numbers p and q are called twin primes i

Problem 1. Two prime numbers p and q are called twin primes if q = p + 2. Prove that there exists an integer a such that p|(a2 q) if and only if there exists an integer b such that q|(b2 p).

Remark: It is a famous open problem to prove that there are infinitely many twin primes.

Solution

Twin primes are prime number which is differ by two,that is either less than 2 or grater than 2.

example:{(3,5) (5,7) (11,13)} these are the twin prime numbers.

here given that two prime numbers p and q.

q=p+2 [here it is also differ by two]

we can written it also as p=q-2

=a such that p mod (a2-q)

==>a2-q=pm [here m is some integer]

= p mod (a2-q)

there exist an integer a such thatt p=pm+q

given that q=p+2 , now substitute q value in equation .

then, p=pm+q

=(q-2)m+q

=qm-2m+q

next q mod (b2-p)

==>b2-p=qm

q mod(b2-q)

there exist an integer b such that

q=qm+p

we substitute q=p+2

then, q=(p+2)m+p

=pm+2m+p