Question

A prime number $p$ is called special if there exist primes $p_{1},\, p_{2},\, p_{3},\, p_{4}$ such that $p = p_{1} + p_{2} = p_{3} - p_{4.}$ The number of special primes is

• A. 0
• B. 1
• C. more than one but finite
• D. infinite

Answer 1

Answer: (B)

It is given that for prime numbers $p_{1}, p_{2} , p_{3}, p_{4}$ the special prime
number

$p=p_{1}+p_{2}=p_{3}-p_{4}$

Case I

If all $p_{1}, p_{2}, p_{3} , p_{4}$ are odd, then $(p_{1} + p_{2})$

and $(P_{3}-P_{4})$ are even, which is not possible

Case II
If one of $p_{1}$ and $p_{2}$ is even, say $p_{2}$ is 2 and $p_{4}$ must be 2.

So, $p = p_{1} + 2 = p_{3} - 2$

the above equation is satisfied only if

$p=5, p_{1}3$ and $p_{3}=7$

So, the number of special prime p is $1$