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What is the rule which tests whether a prime number can be written as the sum of two different squares?
It is Fermat's theorem on the sum of two squares. An odd prime p can be expressed as a sum of two different squares if and only if p = 1 mod(4)
Is 31 a compositeprimeor square number?
31 is a prime number. Please note that all perfect squares are composite, except 0 and 1.
Can a perfect number also be a prime number and how?
A perfect number cannot be a prime.
Is there a product of two perfect squares that is not a perfect square?
No there isn't. every perfect square number can be factored into prime number. At their factoration you'll always have multiples of two on the primes exponent. Therefore you'll multiply a prime raised to a 2-multiple number with another prime raised to a 2-multiple number wich gives you also a number that factored gives you a product of prime numbers raised to a 2-multiple number and so, a perfect square.
What is the number that it is impossible to have 3 divisors for?
Any prime number - by definition such numbers are divisible by 1 and the number itself. As a matter of fact, only perfect squares can have an odd number of factors.
