An algebra problem by Victor Loh

Algebra Level 3

Victor knows that 23=8>22=42^{3} = 8 > 2^{2} = 4 33=27>23=83^{3} = 27 > 2^{3} = 8 43=64>24=164^{3} = 64 > 2^{4} = 16 Since 4324=48>3323=19>2322=4,4^{3} - 2^{4} = 48 > 3^{3} - 2^{3} = 19 > 2^{3} - 2^{2} = 4, Victor concludes that for all integers n2n \geq 2, n3>2n.n^{3} > 2^{n}. However, his math teacher tells him that this only holds true up till a certain value of nn, that is, when nan \geq a where aa is a positive integer, n32n.n^{3} \leq 2^{n}. Find the value of aa.

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