Show that 2 3 is the only pair p1 p2 of primes such that p2

Show that (2, 3) is the only pair (p_1, p_2) of primes such that p_2 = p_1 + 1. A pair of primes (p_1, p_2) is a twin prime pair if p_2 = p_1 +2. Show that every twin prime pair except (3, 5) is of the form (6n - 1, 6n + 1). Show that (2, 5) is the only pair (p_1, p_2) of primes such that p_2 = p_1 + 3. Write down a few pairs (p_1, p_2) of primes such that p_2= p_1 + 4.

Solution

A prime number is a number which is divisible by 1 and itself.

A prime number has only two factors 1 and itself.

(a) (2 , 3) is the only prime ( p1 , p2 ) of primes such that p2 = p1 + 1

If we see the list of prime numbers from 1 to 1000 there are no adjacent numbers which are prime numbers except 2 and 3.

so for p1 = 2 and p2 =3 , 3 = 2+1

so the equation p2 = p1 +1 only saifies for prime numbers (2, 3).

(b) (p1 , p2) is a twin prime pair if p2 = p1 +2

let us take two twin prime pair for p1 = 5 and p2 = 7 (5,7)

( 6n -1, 6n +1 ) = (6*5 - 1 , 6*7 +1 )

=( 29, 41)

the possible remainders of N upon division by 6 is 0, 1,2,3,4,5.

for pair (29, 41 ) the remainders are 1 and 5 when divided by 6.

(c) (p1, p2) of primes such that p2 = p1 + 3

if we take prime numbers between 1 to 50 : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

from above no two adjacent/twin primes have difference 3 between them except (2, 5)

there fore, (2 , 5 ) is the only pair (p1, p2) of primes such that p2 = p1 +3 =5= 2 +3

(d)the few pairs (p1, p2 ) of primes such that p2 = p1 +4

are : (7 , 11), (19, 23 ) , ( 37, 41) , (43, 47 ), (67, 71), (79, 83),....