Number Theory
# Quadratic Diophantine Equations

Magician Mike Michael claims to know two positive whole numbers that multiply to 1000, neither of which contain the digit 0.

What is the sum of these 2 numbers?

\[\large a!b! = a! + b! \]

If \(a\) and \(b\) are positive integers that satisfy the equation above, find \(a + b\).

If we mistakenly add numerator and denominators, we would think:

\[\Large \dfrac{1}{a} + \dfrac{1}{b}=\dfrac{2}{a+b}.\]

How many ordered pairs \((a.b)\) in the interval \(-10\leq a,b \leq 10\) such that they satisfy the equation above.

How many primes are 4 less than a perfect square?

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