Let p and q be natural numbers such that q > p. What is the largest value of p such that q^2 - 5p - 4 is negative?

Explanation

For the largest p, q should be as small as possible, so let q = p + 1. Substitute into the expression: (p+1)^2 - 5p - 4 < 0 \Rightarrow p^2 + 2p + 1 - 5p - 4 < 0 \Rightarrow p^2 - 3p - 3 < 0. Testing values: if p=4, 16-12-3 = 1 > 0 (invalid). If p=3, 9-9-3 = -3 < 0 (valid). So the largest p is 3.

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