1 is not considered a prime number; 1 is a perfect square.
Nice question! The square root of (any number that isn't a perfect square) is irrational. No prime number is a perfect square. So the square root of any prime number is irrational.
3 is prime, 1 less than the perfect square of 4 (2x2)
because it needs to be a prime number and it has to be simplified
No, it's a perfect square 73 * 73.
That is correct.
True. By definition, a prime number is divisible by one and by itself. Also by definition, a perfect square has at least an additional pair of factors - it's square root. Therefore a prime number could never be a perfect square. One exception that might come to mind in this case is the number one. One however, is not considered a prime number, and thus does not conflict with this rule.
4, 16, 64 and many others
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.
A [perfect] square number, by definition, has a factor which is its square root. As a result it CANNOT be a prime!
A perfect number cannot be a prime.
If you mean 'prime' factors, then an odd number of them CAN'T produce a perfect square. Consider 3 factors ... A, B, and C. If their product were a perfect square, then AxB=C, which can't be true if C is a prime number. For non-prime factors, an odd number of them may or may not be a perfect square: Three factors = not a square: 2 x 3 x 5 = 30. Three factors = a square: 2 x 3 x 6 = 36.
The number is 16.
Prime numbers have two factors. All square numbers (other than 1) have more than that.
a perfect square
No. Look at the prime factors of 65: there are no repeated prime factors, therefore this number contains no perfect square.
49 x 14 x 18 is not a perfect square. This is because a prime factorization of 49 x 14 x 18 contains 3 sevens. The prime factorization of any perfect square must have an even amount of each prime number.
17 is a prime number. Its only factors are 1 and 7. Since 1 is a perfect square, it is the answer to the question.
One way is to get the prime factorization of the number. If every prime occurs an even number of times, it is a square, otherwise, not. Another is to estimate the square root of the number, and square it. If you get more than the number, try a lower estimate; if less, a higher one. Using interval bisection you very quickly zero in on the square root, if it is a whole number. If so, the number is a perfect square. Otherwise, you find 2 consecutive whole numbers between which is the square root, in which case it is not a perfect square.
No square number can be a prime number and conversely.