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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.
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.
Can a perfect number also be a prime number and how?
A perfect number cannot be a prime.
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.
Are prime numbers more or less common than perfect squares?
Any number squared except 0 is a perfect square so it follows that prime numbers are less common than perfect squares.
What is the smallest prime number which is the sum of two distinct positive perfect squares?
5
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.
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.
Can a perfect number also be a prime number and how?
A perfect number cannot be a prime.
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.
What occurs when the composite number is expressed as a number of factors that are all prime?
When a composite number is expressed as a product of its prime factors, the prime factorization of the composite number has been performed.
Is 7 a square?
No 7 is not a square number but it is a prime number because it has only two factors which are itself and one.
What is the smallest negative perfect square that is divisible by the four smallest prime numbers?
Perfect squares are positive. A smallest negative number doesn't exist. The four smallest prime numbers are 2, 3, 5 and 7. The smallest perfect square would have to be 2^2 x 3^2 x 5^2 x 7^2 or 44,100
When a number is expressed as the product of prime factors you have?
the prime factorization.
Is one a perfect number?
no, but I don't know why. One is not a prime number and by definition of a perfect number one must be a prime number.
