A Lucas Conjecture

Number Theory Level 5

In a distant empire, quarry workers are given a choice of multiple square boxes with various integer lengths and a height of 1 to pack cubes of granite, each of volume 1, into.

They are taking these blocks to the capital city of Sacul, where they will construct a pyramidal monument for the emperor Edouard II.

The pyramid is constructed with a square base\(^{*}\) such that the center of each cube above aligns vertically with each point that four cubes meet beneath. Every block of granite must be used.

Given the above conditions, what is the maximum length of square box the quarry workers can choose?

\(^{*}\)Note: The pyramid should look something like this:


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