Let N=d
1d
2d
3...d
n be an n-digit decimal number, with n>1.
Form the sum:
S(N) = d1n + d2n+ d3n + ... + dnn
Prove that there are only a finite number of integers N for which S(N)=N.
For an extra credit, find these values of N.
The largest S(N) for a given n would be given by n*9^n.
For any N we must have N >= 10^(n-1)
If first equation grows fast at first but eventually the second will overtake it. I used a table to see happens at n=61. By graphing it's more like 60.868 and you can also use the W function to get an exact form. I used wolfram alpha so see it.
The point is, there's an upper bound.
I first learned of these numbers my freshman year of college (1990) from The Penguin Dictionary of Curious and Interesting Numbers. When I had to take a programming course for prospective high school math teachers I decided to make a lesson out of finding more such numbers using computer programs.
Interestingly, I just found that Armstrong had started using them for the same purpose in the mid 1960's.
These days I rarely use a program to solve things. I leave it to Larry and Charlie.
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Posted by Jer
on 2024-03-15 14:23:43 |
Proof of finiteness and commentary
