Question

Let $x$ and $y$ be two positive integers and $p$ be a prime number. If $x(x - p) - y(y + p) = 7p$, what will be the minimum value of $x - y$?

Accepted Answer

E. Expand equation: $x^2 - px - y^2 - yp = 7p \implies (x^2 - y^2) - p(x + y) = 7p \implies (x - y - p)(x + y) = 7p$. Since $x, y$ are positive integers, $x+y > x-y-p$. Case 1: $x - y - p = 1, x + y = 7p$. Case 2: $x - y - p = p, x + y = 7$. Case 3: $x - y - p = 7, x + y = p$. Checking integer conditions yields no consistent positive integer solution for $x, y$ with $p$ prime.

Integer Solutions to Prime Equation

Explanation

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