Consider what prime means from a visual point of view - not having any factors except itself and 1 means if you are arranging a prime number of tiles in a rectangular grid you can only make a rectangle of dimensions 1 by (prime). That's what it means to be prime. read more
If a number is prime, that mean it has only two divisor (1 and itself). A perfect square number n can be written as m*m (where m = square_root(n)) or in other words, n = m*m which means m is also divisor of n and it contradicts definition of prime number. So there can no prime number which is also a perfect square. read more
A perfect square is a number that can be expressed as the product of two equal integers. Since the only factors of any prime is itself and 1, no prime can be a perfex\ct square. Definition: A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. read more
Find all primes $p$ and $q$ such that $p^2 + 7pq + q^2$ is a perfect square. One obvious solution is $p = q$ and under such a situation all primes $p$ and $q$ will satisfy. Further if $p \neq q$ then we can assume without the loss of generality that $p > q$. read more