If the greatest common divisor (GCD) of a , b is 1, c, Verify a-b is a perfect square such that \dfrac{ab}{a-b} =c

If m is a four-digit positive number, show there exists positive number n such that m-n is prime and mn is a perfect square number.

If n is positive integer greater than 1913, find the value of n such that \dfrac{n-1913}{2013-n} is a perfect square.

Let p >2 be a prime. Prove that there exists at least one positive integer n such that n^2+np is a perfect square.