If p is prime, then p^2 +p + 1 is also prime.
The statement "if p is prime, then p^2 +p+1 is also prime" is actually a known proposition. This is a special case related to the class of prime numbers known as "Sophie Germain primes." A Sophie Germain prime is a prime number p for which 2p+1 is also prime.
In the given expression p^2 +p+1, if p is prime, then this expression represents a quadratic form associated with the Sophie Germain identity, which is a factorization:
p^2 +p+1=(p+1)^2 −p
This form suggests that p^2 +p+1 can be expressed as the square of a binomial (p+1) minus p. If p is prime, then both p+1 and p are consecutive integers, and the expression (p+1)^2 −p may be prime under certain conditions.
However, it's important to note that the proposition does not hold for all primes; it's a specific characteristic related to Sophie Germain primes. While Sophie Germain primes exhibit this property, not all primes follow this pattern. As a result, this statement is a conditional one and should be applied selectively to Sophie Germain primes rather than to all prime numbers.