When I was in school, perhaps my favorite pet project was trying to find a formula for the number of digits in . For starters:
: 1 digit
: 1 digit
: 1 digit
: 1 digit
: 2 digits
: 3 digits
: 3 digits
: 4 digits
: 5 digits
I owned what was then a top-of-the-line scientific calculator (with approximately the same computational capability as a modern TI-30), and I distinctly remember making a graph like the following on graph paper. The above calculations contribute the points ,
,
,
,
,
,
,
, and
.
I had to stop (or, more accurately, I thought I had to stop) at
because my calculator couldn’t handle numbers larger than
.
I stared at this graph for weeks, if not months, trying to figure out an equation that would fit these points. And I never could figure it out.
And, to this day, I’m somewhat annoyed at my adolescent self that I wasn’t able to figure out this puzzle for myself… since I had all the tools in my possession needed to solve the puzzle, though I didn’t know how to use the tools.
In this series of posts, I’ll answer this question with the clever application of some concepts from calculus and precalculus.
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