Out of all 2-digit strings, 68 is the last to appear in the decimal expansion of pi. ]]>

I misread this as some sort of nonsense joke about the “last 2-digit string to appear in pi” before realizing you meant “68 is the two-digit string that initially appears in the decimal expansion of pi farthest from the start”.

]]>It would be interesting to divide all your properties into 2 classes: those which are true in all representations, and those which are dependent on decimal (or some other) representation. The former feel like properties of the number, the latter more as properties of the string of digits.

Of course, the REALLY interesting case would be to find numbers with properties which are about the representation but held for ALL bases. For example, if the property of 68 held in all bases: something like, 68 is the largest number N such that N, represented in any base B (taking m digits) is the last m-digit string of digits in the base-B representation of pi, for all B.

Of course, then I would ask you whether the property held in bases which are not successive powers of a single number (that has been an interest of mine since I found out, in my first job in 1978, that APL included a simple operator that could convert a value to a vector of numbers representing the value in a base formed from arbitrary numbers – it was designed for things like converting weights to stones, pounds and ounces but seemed like an interesting area for research).

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