Question

A prime number p is called special if there exist primes p₁, p₂, p₃, p₄ such that p = p₁ + p₂ = p₃ - p₄. The number of special primes is

(a) 0

(b) 1

(c) more than one but finite

(d) infinite

Answer 1

Answer is : (b) 1

It is given that for prime numbers p₁, p₂, p₃, p₄ the special prime number

p = p₁ + p₂ = p₃ - p₄

Case I

If all p₁, p₂, p₃, p₄ are odd, then (p₁ + p₂ ) and (p₃ - p₄) are even, which is not possible.

Case II

If one of p₁ and p₂ is even, say p₂ is 2 and p₄ must be 2.

So, p = p₁ + 2 = p₃ - 2

the above equation is satisfied only if

p = 5, p₁ = 3 and p₃ = 7

So, the number of special prime p is 1.