Problem 5

Determine the largest integer n with the property that n is divisible by every positive integer less than the cube root of n.

Step 4 of 4: Contradict the bound for t at least 13

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t^4\le 12t^3+7t^2+24t+6,\qquad 1\le\frac{12}{t}+\frac7{t^2}+\frac{24}{t^3}+\frac6{t^4}<1

Detailed analysis

Combining the preceding inequality with N<(t+1)^3 and expanding gives t^4 <= 12t^3+7t^2+24t+6. After division by t^4 this says 1 <= 12/t+7/t^2+24/t^3+6/t^4. But for t>=13 the right side is less than 1, a contradiction. Thus no N>420 works, and the largest integer is 420.