I searched numbers $N$, such that the continued fraction of $N^{1/3}$ has very large entries. I only searched for a single large entry, but I was surprised that two continued fractions contained not only one large entry, but multiple large entries. Here the two amazing expansions :
$$102175 [46, 1, 2, 1, 8741, 2, 186, 2, 13112, 1, 6, 1, 8, 2, 9, 2, 623, 1, 33, 1 , 9, 1, 2, 2, 17484, 14, 2, 2, 1, 4, 19021, 2, 1, 1, 1, 1, 1, 1, 3437888, 2, 2, 6, 21510, 2, 1, 2, 55063048, 1, 1, 1, 1, 1, 2, 8, 44, 2, 1, 4, 1, 4, 61, 2, 1666 1, 2, 1, 3, 1, 1, 23, 1, 4, 2, 2, 8, 3, 3, 1, 1, 2, 6, 3, 1, 1, 3, 5, 17, 21, 17 , 3, 168, 3, 1, 1, 17, 1, 3, 2, 3, 4, 3] 55063048$$ $$267090 [64, 2, 2, 31104, 1, 4, 64, 4, 1, 46657, 1288, 55545, 1127, 62210, 2, 2, 40, 1, 1, 2, 1, 1, 4, 559, 8, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 101091, 3, 1, 1, 8, 6, 10, 3, 1, 2, 2, 1, 1, 2, 17, 2, 1, 1, 2, 1, 4897902700, 1, 54, 1, 288, 1, 1, 1, 20, 1, 1, 5, 31360929, 1, 15, 9, 1, 1, 30, 1, 5, 6, 2, 7, 16, 2, 3, 1, 2, 3, 9935, 1, 3, 2, 1, 5, 4, 4, 2, 1, 28, 1, 27] 4897902700$$
Explanation : First, the number $N$ is displayed, then the first $100$ terms of the continued fraction of $N^{1/3}$ and finally the maximum of the entries.
The cubic roots of the numbers $102175$ and $267090$ seem to have a very special continued fraction.
In the second continued fraction, we have even $5$ consecutive large entries, and in both continued fractions we have an entry larger than $10^6$ besides the maximum entry.
This is not at all what I expected, in particular because almost every real number has a continued fraction expansion that follows a special distribution mentioned here :
Typicality of boundedness of entries of continued fraction representations
The continued fraction expansions above are far away from this distribution (even if we do not consider the maximum entry).
How can this phenomen be explained ?