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Quadratic Diophantine Equations

The Pythagorean Theorem is one of the most famous quadratic diophantine equations, which are equations with squared variables solved over the integers. How many solutions can you find?

Level 2

         

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.

What is the smallest positive multiple of hundred which can be expressed as the product of two consecutive integers?

How many primes are 4 less than a perfect square?

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