Prime numbers p with expression p = a2 ± ab ± b2

• For any pair of integers m and n  , define Km,n:={a2+mab+nb2|a,b∈Z}Km,n:={a2+mab+nb2|a,b∈Z}.
• Km,nKm,n is a semi-group with usual product of integers, for any pair of integers m and n.
• A prime number p   can be expressed as p=a2±ab−b2p=a2±ab−b2 with integers a and b, if and only if, p is congruent to 0, 1 and −1 modulo 5.
• A prime number p   can be expressed as p=a2±ab+b2p=a2±ab+b2 with integers a and b, if and only if, p is congruent to 0 and 1 modulo 3.

Let p be a prime number. In this paper we show that p   can be expressed as p=a2±ab−b2p=a2±ab−b2 with integers a and b if and only if p   is congruent to 0, 1 or −1 (mod5) and p   can be expressed as p=a2±ab+b2p=a2±ab+b2 with integers a and b if and only if p   is congruent to 0, 1 (mod3).